This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
0, &2 > 0, and f in L'2(D} be given such that minjfci, fc2} > & for some constant a > 0. For any sequence {Xn} and x in coX(D) such that Xn —» X € L2(D)-strong, there exists a subsequence {ynk = y(Xnk}} of {yn = y(Xn}} such that
and any strongly convergent subsequence of {yn} converges to the same limit. Proof. For each n, let kn = kiXn + ^2(1 — Xn} and let yn 6 HQ(D) be the solution of the equation
The difference zn = yn — y is the solution of the variational equation
By assumption, kn > a. Setting (p = zn, we get
7. Continuity of the Dirichlet Boundary Value Problem
149
If AI = Ai(-D), the first eigenvalue of the Laplacian —A in HQ(D], then
On the right-hand side we use the Lebesgue-dominated convergence theorem as follows: there exist some subsequences {knk — k} (resp., {Xnk — %}) which converge to 0 almost everywhere in D while at the same time
The last part of the lemma is straightforward. 7.2
Approximation by Transmission Problems
We now show that, as e —» 0, the convergence in HQ(D) of ye(x] to the solution y ( x ) of the relaxed homogeneous Dirichlet problem (7.1)-(7.2) can be controlled through the norm of some extension operator. Theorem 7.1. Let D be a bounded open connected domain in RN and 0 a measurable subset of D such that
Assume that there exists a continuous linear extension operator
where \\P\\ is the norm of P in C(H(£IC},HQ(D}}. Given x = Xjy ande, 0 < £ < 1, let ye be the solution of the variational equation (7.5) in HQ(D) and y the solution of the variational equation (7.1) in the space Hl(£l; D) defined in (7.2) for x = XnThen there exists a constant c'(D] > 0 such that
Proof. For e, 0 < £ < 1, substitute (p = y£ in (7.5) to get the first bound
150
Chapters. Relaxation to Measurable Domains
In view of the properties of f), ||V 0 such that for all n,
Then if Xfin ~^ Xn y(Xa )} such that
?n
L2(D]-strong, there exists a subsequence {ynk} of {yn =
7. Continuity of the Dirichlet Boundary Value Problem
151
//, in addition, all the domains are Lipschitzian, then if Xnn ~^ Xn ^n L2(D)-strong, there exists a subsequence {ynk} of [yn = y(xsir,}} sucn that
Proof. The proof follows from Proposition 7.1 and Theorem 7.1.
Remark 7.1.
In Agmon, Douglis, and Nirenberg [1, 2] the extension operator is constructed from the locally uniform cone property of Definition 6.1 in Chapter 2. It can be verified that for a family of domains 0 satisfying the uniform cone property in a bounded hold-all D, the corresponding extension operators P^ and P^c are uniformly bounded with respect to all such domains 0 in D.
Remark 7.2.
In section 7 of Chapter 6 a sharper continuity of the homogeneous Dirichlet problem for the Laplacian will be given under capacity conditions. Nevertheless, the penalization technique used in this section can be applied to higher-order elliptic, and even parabolic, problems and the Navier-Stokes equation (cf. Dziri and Zolesio [6]).
This page intentionally left blank
Chapter 4
bpologies Generated by bistance Functions
1 Introduction In Chapter 3 the characteristic function was used to embed the equivalence classes of measurable subsets of D into LP(D] or Lf oc (D), 1 < p < oo, and induce a metric on the equivalence classes of measurable sets. This construction is generic and extends to other set-dependent functions embedded in an appropriate function space. The Hausdorff metric is the result of such a construction, where the distance function plays the role of the characteristic function. The distance function embeds equivalence classes of subsets A of a closed hold-all D with the same closure A into the space C(D) of continuous functions. When D is bounded the C*°-norm of the difference of two distance functions is the Hausdorff metric. The Hausdorff topology has many much-desired properties. In particular, for D bounded the set of equivalence classes of nonempty subsets A of D is compact. Yet the volume and perimeter are not continuous with respect to the Hausdorff topology. This can be fixed by changing the space C(D] for the space Wl'p(D] since distance functions also belong to that space. With that metric the volume is again continuous. The price to pay is the loss of compactness even when D is bounded. But new sequentially compact subfamilies can easily be constructed. By analogy with Caccioppoli sets we introduce the sets for which the elements of the Hessian matrix of second-order derivatives of the distance function are bounded measures. They are called sets of bounded curvature. Their closure is a Caccioppoli set, and they seem to enjoy other interesting properties. Convex sets belong to that family. General compactness theorems are obtained for such families under global or local conditions. This chapter simultaneously deals with the family of open sets characterized by the distance function to their complement. They are discussed in parallel with the sets described by their distance function. The properties of distance functions and Hausdorff and Hausdorff complementary metric topologies are studied in section 2. The differentiability of distance functions is discussed in section 3 along with the notions of projections, skeleton, and cracks. W^'P-topologies are introduced in section 4. The compact families of 153
154
Chapter 4. Topologies Generated by Distance Functions
sets of bounded and locally bounded curvature are characterized in section 5. The special families of convex sets and Federer's sets of positive reach are studied in sections 6 and 7 and will be further investigated in Chapter 5. Finally, section 8 gives several compactness theorems under global and local conditions on the Hessian matrix of the distance function.
2
Hausdorff Metric Topologies
2.1
The Family of Distance Functions Cd(D]
In this section we review some properties of distance functions and present the general approach that will be followed in this chapter. Definition 2.1. Given A C RN the distance function from a point x to A is defined as
and the family of all distance functions of nonempty subsets of D, as
When D = R N the family C d (R N ) is denoted Cd. Observe that dA is finite in RN if and only if A ^ 0. We recall the following properties of distance functions. Theorem 2.1. Assume that A and B are nonempty subsets o/R N . (i) The map x i—>• dA(x) is uniformly Lipschitz continuous in R N 7
anddAeC^lc(W).1 (ii) There exists y € A such that dA(x] = \y — x and dA = d-^ in R N .
1
A function / belongs to C^(R N ) if for all bounded open subsets D of RN its restriction to D belongs to C°>l(D).
2. Hausdorff Metric Topologies (vii) d& is (Frechet) differentiable
155 almost everywhere and
Proof, (i) For all z 6 A and x, y G RN
and hence the Lipschitz continuity. (ii) Let {yn} C A be a minimizing sequence
Then {yn — x} and hence {yn} are bounded sequences. Hence there exist a subsequence converging to some y G A and cU(x) = |y — £ • Clearly by definition d^(x) < d/i(x) since A C A. So either there exists y G A such that c^(x) = \y — x and hence d-^(x) = ^(x) or there exists {yn} C A such that
and we get the identity. By the continuity of d^, d~^{ti} is closed and
Conversely, if cU(x) = 0, there exists y in A such that 0 = d^(x) = \y ~- x and necessarily x = y G A. (iii) follows from (ii). (iv) follows from (iii). (v) follows from (ii) for x G A, C^A(X) = 0, and ds(x) < d^(^) = 0 necessarily implies d#(x) — 0 and x e B. Conversely, if A C B, then
(vi) is obvious, (vii) follows from Rademacher's theorem. 2.2
Hausdorff Metric Topology
Let D be a nonempty subset of RN and associate with each nonempty subset A of D the equivalence class
since from Theorem 2.1(v) d& = ds if and only if A = B. representative of the class [A]. Consider the set
So A is the closed
156
Chapter 4. Topologies Generated by Distance Functions
By the definition of [A] the map
is injective. So ^(D] can be identified with the subset of distance functions Cd(D] in C(D). The distance function plays the same role as the set X(£>) in LP(D] of equivalence classes of characteristic functions of measurable sets. When D is bounded, C(D] is a Banach space when endowed with the norm
As for the characteristic functions of Chapter 3, this will induce a complete metric
on F(D), which turns out to be equal to the classical Hausdorff metric
(cf. Dugundji [1, p. 205, Chap. IX, Prob. 4.8] for the definition of px). When D is open but not necessarily bounded, the space C(D) of continuous functions on D is endowed with the Frechet topology of uniform convergence on compact subsets K of D, which is defined by the family of seminorms
It is metrizable since the topology induced by the family of seminorms {QK} is equivalent to the one generated by the subfamily {qKk}k>ii where the compact sets {Kf-}k>i are chosen as follows:
(cf., for instance, Horvath [1, Example 3, p. 116]). Thus the Frechet topology on C(D] is equivalent to the topology defined by the metric
In that case we use the notation C\OC(D). It will be shown below that Cd(D] is a closed subset of C\OC(D] and that this will induce the following complete metric on^(D):
which is a natural extension of the Hausdorff metric to an unbounded domain D.
2. Hausdorff Metric Topologies
157
Theorem 2.2. Let D be a nonempty open (resp., bounded open) subset o/R N . (i) The set Cd(D) is closed in C\OC(D) (resp., C(D)) and p$ (resp., p) defines a complete metric topology on J~(D). (ii) When D is a bounded open subset o/R N , the set Cd(D) is compact in C(D) and the metrics p and pn are equal. Proof, (i) We use the proof given by Dellacherie [1, p. 42, Thrn. 2 and p. 43, Remark I}. The basic constructions will again be used in Chapter 5. Consider a sequence {An} of nonempty subsets of D such that d,An converges to some element / of C\OC(D). We wish to prove that / = dA for
and that the closed subset A of D is nonempty. Fix x in D; then
Hence {yn — x} and {yn} C D are bounded, and there exists a subsequence, still indexed by n, which converges to some y 6 D,
In particular, f(y) = 0, since in the inequality
the last term is zero and d^n is Lipschitz continuous with constant equal to 1:
and both terms go to zero. By the definition of A, y £ A and A is not empty. Therefore for each a: € D, there exists y £ A such that
Next we prove the inequality in the other direction. By construction for any An C D
and
By uniform convergence the first and last terms converge to zero, and by Lipschitz continuity of d^n,
158
Chapter 4. Topologies Generated by Distance Functions
Hence for all x E D and y G A, f(y) = 0 and
This proves the reverse equality. (ii) Observe that on a compact set D and for any A C D
The compactness of Cd(D) now follows by the Ascoli-Arzela Theorem 2.1 of Chapter 2 and part (i). By construction p and ps are metrics on F(D}. By definition, for A and B in the compact D,
Conversely, for any x G D and y in A,
and there exists XB G -B such that d,B(x) = \ X - X B \ - Therefore,
Similarly and for all x in D
When D is bounded the family F(D] enjoys many more interesting properties. Theorem 2.3. Let D be a nonempty open (resp., bounded open) subset of R N . Define for a subset S of RN the sets
2. Hausdorff Metric Topologies
159
(i) Let S be a subset o/R N . Then H(S) is closed in Cioc(D) (resp., C(D}). (ii) Let S be a closed subset o/R N . Then I(S) is closed in C\OC(D) (resp., C(D}}. If, in addition, STlD is compact, then J ( S ) is closed in C\OC(D] (resp., C(D}}. (\\\) Let S be an open subset o/R N . Then J ( S ) is open in C\OC(D) (resp., C(D}). If, in addition, CtSH-D is compact, then I(S) is open in C\OC(D) (resp., C(D)). (iv) For D bounded, associate with an equivalence class [A] #([A]} — number of connected components of A. Then the map is lower semicontinuous. In particular, for a fixed number c > 0 the subset
is compact in C(D}. Proof, (i) If S &A- So for any sequence {dAn} in H(S) converging to &A in C\OC(D),
(ii) We use the same technique for I ( S ) as for H(S}, but here we need S = S to conclude. For D fl S = 0, J(S) = 0 and there is nothing to prove. Assume that D n S ^ 0 and consider a sequence {d^n} in J(S) converging to &A in Cioc(D). Assume that ~A H__S = 0. Then ~A C D implies A n [5 n D] = ~A n S = 0. By assumption, S Pi D is compact and
So for all x e D n 5
and this contradicts the fact that An e J(S). (iii) By definition, for any 5, CJ(5) = 7(05):
Since CS1 is closed, CJ(5) is closed from part (ii) and J(S] is open. Similarly, replacing S by C51 in the previous identity, 7(5) — CJ(C5). So from part (ii) if D n C5 is compact, then C/(5) is closed and I(S) is open. From part (ii) if D D C5 is compact, then C/(5) is closed and 1(5) is open.
160
Chapter 4. Topologies Generated by Distance Functions
(iv) Let {An} and A be nonempty subsets of D such that d,An converges to d^ in C(D}. Assume that #([-4]) = k is finite. Then there exists a family of disjoint open sets GI , . . . , Gk such that
In view of the definitions of I(S) and J(S) in part (i),
But U is not empty and open as the finite intersection of k -f 1 open sets. As a result there exists e > 0 and an open neighborhood of [A],
Hence since d,An converges to c^, there exists n > 0 such that
and necessarily
Therefore, [A] i—>• #([A]) is lower semicontinuous. Now for #([A|) = +00, we repeat the above procedure and refine the open covering. This theorem has many interesting corollaries. For instance, part (iii) says something about the function that gives the number of connected components of A (cf. Richardson [1] for an application to image segmentation).
2.3
Hausdorff Complementary Metric Topology and C%(D)
In the previous section we dealt with a theory of closed sets, since the equivalence class of the subsets of D was completely determined by their unique closure. In partial differential equations the underlying domain is usually open. To accommodate this point of view, consider the set of open subsets 0 of a fixed nonempty open hold-all D in R N , endowed with the Hausdorff topology generated by the distance functions d^ to the complement of 0. This approach has been used in several contexts, for instance, in Zolesio [6, section 1.3, p. 405] in the context of free boundary problems. Also, Sverak [2] uses the family C%(D) for open sets 17 such that #([Cfi]) < c for some fixed c > 0. His main result is that in dimension 2 the convergence of a sequence {On} to fi of such sets implies the convergence of the corresponding projection operators {P$in : HQ(D) —» #0(^1)} to P^: HQ(D] —> #g(Q), where the projection operators are directly related to homogeneous Dirichlet linear boundary value problems on the corresponding domains {f2n} and fl In dimension 1 the constraint on the number of components can be dropped. This result will be discussed in Theorem 8.2 of section 8 in Chapter 6.
2. Hausdorff Metric Topologies
161
By analogy with the constructions of the previous section, consider for a nonempty subset D of RN the family
By definition,
and, by Lipschitz continuity,
and in C0(int D) if D is bounded. If int D = 0, then dCA = 0 in R N . If int D ^ 0, associate with each A the open set
By definition and the previous considerations
So finally, for int D ^ 0,
From this analysis it will be sufficient to consider the family of open subsets of an open hold-all.
Definition 2.2.
Let D be a, nonempty open subset of R N . Define the family of functions
corresponding to the family of open sets
For D bounded define the Hausdorff
complementary metric p°H on G(D):
Note that for QI C D and f^ C D, d^l — d^2 — 0 in CD. Therefore,
162
Chapter 4. Topologies Generated by Distance Functions
Theorem 2.4. Let D be a nonempty open subset o/R N . (i) The set Ccd(D] is closed in Cioc(D}. (ii) //, in addition, D is bounded, then C^(D] is compact in Co(D) and (G(D},p°H} is a compact metric space. (iii) (Compact!vorous property) Let {fin} and ft, be sets in Q(D] such as
Then for any compact subset K C ft, there exists an integer N(K) > 0 such that Proof, (i) Let {On} be a sequence of open subsets in D such that {d^n} is a Cauchy sequence in C\OC(D}. For all n, d^n = 0 in CD and {d^n} is also a Cauchy sequence in (7ioc(R-N)- By Theorem 2.2 (i) the sequence {d^n} converges in CIOC(RN) to some distance function d&. By construction
By choosing the open set Jl = \jA in D we get
(ii) To prove the compactness we use the compactness of Cd(D] from Theorem 2.2 (ii) and the fact that C%(D) is closed in Co(D). Observe that since Cnn D CD, d^n = 0 in CD, d^n £ C0(D), and
There exists a subsequence, still indexed by n, and a set A, 0 ^ A C D such that
since d^n = 0 = 0 since x E K C ft and y E $£7. Now
Notation 2.1. It will be convenient to write for the Hausdorff complementary convergence of open sets of G(D],
3
Projection, Skeleton, Crack, and Differentiability
In this section we study the connection between the gradient of dA and d^A and the projections and the characteristic functions associated with A and CA. We further relate the set of singularities of the gradients and the notions of skeleton and set of cracks.
Definition 3.1. (i) Given A C R N , 0 ^ A (resp., 0 ^ CA), and x E R N , the set of projections of x on A (resp., CA) is given by
The elements of n^(x) (resp., I\.^A(x]} are called projections onto A (resp., CA) and denoted by PA(X] (resp., PCA(X}}(ii) Given A C R N , 0 ^ A (resp., 0 ^ CA), the set of points where the projection on A (resp., CA) is not unique,
is called the exterior (resp., interior) skeleton. Since for x E dA the sets n^(x) and HQA(X) are singletons,
164
Chapter 4. Topologies Generated by Distance Functions
Figure 4.1. Nonuniqueness of the projection and skeleton. Intuitively, the smoothness of the boundary dA of A is related to the smoothness of Vd,A(x) in a small exterior neighborhood of the boundary dA of A. However, even for very smooth sets, Vd^(x] may not exist far from the boundary, as shown in Figure 4.1, where Vd^(x) exists outside A except on a semi-infinite line. The directional derivative of the square of the distance function can be computed by using a theorem on the differentiability of the min with respect to a parameter (cf. Chapter 9, section 2.3, Thm. 2.1). It is related to the support function of the set
ru(z).
Theorem 3.1. Let A, 0 ^ A C R N , and x <E R N . Define
(i) The set HA(X) is nonempty, compact, and
(ii) For all x and v in RN
and coB is the convex hull of B. (ni) The following statements are equivalent:
3. Projection, Skeleton, Crack, and Differentiability
(a) d\(x} is (Frechet) differentiable (b) d\(x] is Gateaux differentiable
165
at x, at x,
(c) n^a?) is a singleton. Henceforth
where
When d\ is differentiable
at x, 11,4(2;) = [p^(x)} is a singleton and
For all x G A, UA(X) = {x}, d\ is differentiable
at x and Vd2A(x) = 0.
(iv) For £>A j^ 0, the conclusions of parts (i)-(iii) are true with £>A in place of A and
in place of C ext (A). Proof, (i) Existence. The function
is continuous and z — x\2 —> OD as \z\ —> oo. Hence for any
the set
is nonempty and compact and
Thus HA(X) is nonempty and
166
Chapter 4. Topologies Generated by Distance Functions
is compact. By definition, UA(X) C A. If x £ dA, PA(X) = £ £ 0. Now if PA(%) £ int.4, then there exists an open ball B of radius r, 0 < r < dA(x)/2, at £u(:r), which is contained in A. Choose
Then y belongs to A and
So there exists y £ A such that
This contradicts the minimality of cU(:r). Hence a; £ 0, and consider for t > 0 the quotient
For all p 6 HA (x) and pt £ 11^ (x + tv]
and for all p £ H^ (x)
In the other direction choose a sequence t^ > 0 such that tk —> 0 and Po- But by continuity
Therefore,
Hence (iii) From part (ii) d\(x) is Gateaux differentiable at x if and only if UA(X) is a singleton. The (Frechet) differentiability is a standard part of Rademacher's theorem, but we reproduce it here for completeness as a lemma. Lemma 3.1. Let f : RN —>• R be a locally Lipschitzian function. (Frechet) differentiable at x; that is,
if and only if it is Gateaux differentiable
and the map v i—>• df(x;v)
Then f is
at x; that is,
is linear and continuous.
Proof. It is sufficient to prove that the Gateaux differentiability implies that
The function / is locally Lipschitzian and there exist 61 > 0 and c\ > 0 such that / is uniformly Lipschitzian in the open ball B(x,5i) with Lipschitz constant c\. Therefore, |V/(x)| < cl. The unit sphere 5(0,1) = {v e RN : \v\ = 1} is compact and can be covered by the family of open balls {B(v, £/(4ci)) : v € 5(0,1)} for an arbitrary e > 0. Hence there exists a finite subcover {B(vn,£/(4ci}) : I < n < N} of 5(0,1). As a result, given any v G 5(0,1), there exists n, 1 < n < N, such that \v — vn < e/(4ci). Define the function
168
Chapter 4. Topologies Generated by Distance Functions
for x G R N , v G 5(0,1), and t > 0. Since f(x] is Gateaux clifferentiable
Hence for \y — x < 6
and we conclude that f ( x ) is differentiable at x. From the above equivalences, when d\ is differentiable at x, then T[^(x) = {PA(X}} is & singleton, and from part (ii)
which yields the expression for PA(X). When x £ A, UA(X) — {x}, and by substitution, Vd^(rr) = 0. This completes the proof of the theorem. Remark 3.1. In general, for each v G R N ,
but the choice of p depends on the direction v and is not necessarily unique. The set-valued map (/C(R N ), the set of nonempty compact subsets of R N ) contains a lot of information about the set A. We now turn to the differentiability of d,A- The distance function is uniformly Lipschitzian of constant 1 and differentiable almost everywhere. When it is differentiable, d\ is also differentiable, and from Theorem 3.1, UA(X] contains a unique element PA (x). However, the smoothness of the boundary dA does not imply that HA(X) is a singleton everywhere in R N . For instance, if
then dA is C°° but H^(0) = A. However, we shall see in section 6 that for convex sets A, HA(X) is always a singleton. We now explicitly compute the gradients of dA and d$A and relate them to the characteristic functions of the closures of A and CA
3. Projection, Skeleton, Crack, and Differentiability
169
Theorem 3.2. (i) Let A, 0 ^ A c R N . // VcU(#) exists at a point x in R N , then H^(x) = {PA(X)} is a singleton,
(ii) For A, 0 ^ A C R N , and x G RN \dA, dA is differentiable d2A is differentiable at x.
at x if and only if
(iii) Given A C R N , 0 ^ A (resp., 0 ^ (M),
and for almost all x
(iv) Given x G R N 7 a G [0,1], p G 11^(re), anrf xa = p + a (x — p ] ,
In particular, i/n^(x) is a singleton, then n^(a;a) ^s a singleton and Vd2A(xa] exists for all a, 0 < a < 1. //, m addition, x ^ A, VdA(xa] exists for all 0 < a < 1. Proof, (i) If VeU(x) exists for x G A, then for all v G RN
which implies that Vd^(^) = 0. If Vd^(x) exists for x ^ A, then
exists, and by Theorem 3.1, Ilyi(x) = {PA(X}} is a singleton and Vc^(x) = 2(x — PA(X})- But for x £ A, and it is sufficient to divide both sides by 2 PA(X] — x .
1 70
Chapter 4. Topologies Generated by Distance Functions
(ii) If VdA(x] exists, then Vd\(x) = 2dA(x) VdA(x) exists. Conversely, asA(x) exists. If x G int A, then dA = 0 in some neighborhood of x and hence VdA(x) exists and is equal to zero. If x ^ A, then for t ^ 0
So VdA(x) exists and is equal to
Vd\(x)/(2dA(x)}.
(iii) From part (i) if VdA(x) exists for x G A, it is equal to 0. Hence X~A(X] — 1 = 1- \Vd exists and |VcU(#)| = 1 for all points in int (L4\Skext (A}. Hence from part (i) XA(x) = 0 = 1 - |VcU(ff)| = 0 on int k4\Skext (-4). Since m(Sing (VoU)) = 0, the above identities are satisfied almost everywhere in R N . We get the same results with CA in place of A. (iv) If x G A, 11,4(0;) = {x} and there is nothing to prove. If x £ A, then dA(x) = \x — p\ > 0 and p ^ x for all p 6 IlA(x). First
If the inequality is strict, then there exists pa 6 A such that xa — pa = dA(xa) and
This contradicts the minimality of dA(x) with respect to A. Therefore, Now for any pa e HA(xa), pa £ A and and pa G n^(o;). Hence TiA(xa} C 11^(0;). This completes the proof. Observe that for points outside of A the norm of VdA(x) is equal to 1. It coincides with the outward unit normal to A at the point p(x] when A is sufficiently smooth. When A is not smooth this normal is not always unique, as can be seen in Figure 4.2. We now complete the description of the singularities of the gradients of dA and dCA,
by giving a name to the sets C ex t (A) and Ci nt (A) introduced in Theorem 3.2.
3. Projection, Skeleton, Crack, and Differentiability
171
Figure 4.2. Nonuniqueness of the exterior normal.
Definition 3.2. (i) For A C R N , A ^ 0, the set of exterior cracks is defined as
(ii) For A C R N , CA 7^ 0, the set of interior cracks is defined as
Points of Cext (A) cannot belong to RN \dA since, from Theorem 3.2 (ii), for such points the existence of Vd2A(x] implies the existence of Vd^(x). The same is true for C int (A). Therefore,
The sets Cext (A) and C int (A} are generally not equal to all of dA, as can be seen from the next example.
Example 3.1. Let £?(0,1) be the open ball of radius 1 at the origin and let
Then
Note that in the equivalence class [A], the boundary can be different for two members of the class. In fact, dA C dA for the closed representative of A and this inclusion can be strict, as shown from this example.
172
4
Chapter 4. Topologies Generated by Distance Functions
W^-Topology and Characteristic Functions
Distance functions are locally Lipschitzian. They belong to Clo'^(RN) and hence to VFlo'^(RN) for all p > I . Thus the previous constructions can be repeated with W^(D] in place of C\OC(D] to generate new HAl>p-metric topologies on the family Cd(D). One big advantage is that the Wl^-convergence of sequences will imply the //^-convergence of the corresponding characteristic functions of the closure of the sets and hence the convergence of volumes (cf. Theorem 3.2 (iii)). The convergence of volumes and perimeters is lost in the Hausdorff topology. The next example shows that the HausdorfF metric convergence is not sufficient to get the Z/p-convergence of the characteristic functions of the closure of the corresponding sets in the sequence. The volume function is only upper semicontinuous with respect to the Hausdorff topology. The perimeters do not converge either. This is also illustrated by the example of the staircase of Example 5.1 and Figure 3.7 in Chapter 3, where the volumes converge but not the perimeters.
Example 4.1.
Denote by D = [—1,2] x [—1,2] the unit square in R2, and for each n > 1 define the sequence of closed sets
Figure 4.3. Vertical stripes of Example 4.1. This defines n vertical stripes of equal width l/2n each distant of l/2n (cf. Figure 4.3). Clearly, for all n > 1,
where S = [0,1] x [0,1] and Pp(An} is the perimeter of An. Hence
4. Wl'p-Jopo\ogy and Characteristic Functions
173
But
Since the characteristic functions do not converge, the sequence {Vc^,} does not converge in LP(D)N. Theorem 4.1. Let D be an open (resp., bounded open) subset o/R N . (i) The topologies induced by W^(D) (resp., Wl>p(D)) on Cd(D) and Ccd(D] are all equivalent for p, I < p < oo. (ii) Cd(D) is closed in W^(D) (resp., Wl'p(D)) for p, 1 < p < oo, and
defines a complete metric structure on F(D}. For p, 1 < p < oo, the map
is "Lipschitz continuous": for all bounded open subsets K of D and nonempty subsets A\ and AP(.D) into Wljl(D) is continuous since
with l/p + l/q — 1. Conversely, we have the continuity in the other direction. For p > 1 and any d& and ds in Cd(D),
Therefore, for any e > 0, pick 6 = £p/(2 maxjc, I})*3"1 and
(ii) It is sufficient to prove it for D bounded open. Let {d^.n } be a Cauchy sequence in Cd(D] which converges to some / in VFlip(.D)-strong. By Theorem 2.2 (ii), Cd(D) is compact in C(D] and there exist a subsequence, still denoted {cUn}, and 0 ^ A C Z> such that and, a fortiori, in Lp(D)-strong since D is bounded. By uniqueness of the limit in Lp(D)-strong, / = d^ and d^n converge to cU in H/1>p(D)-strong. Therefore, Cd(D] is closed in Wljp(D). For the Lipschitz continuity, recall that the distance function dA is differentiate almost everywhere in RN for A ^ 0. In view of Theorem 3.2 (iv)
Given two nonempty subsets AI and A% of D
4. VF1'p-Topology and Characteristic Functions
175
for 1 < p < oo and with the ess-sup norm for p = oo. (iii) Again it is sufficient to prove the result for D bounded. In that case CfJ 7^ 0 for all open subsets £1 of D. Let {On} be a sequence of open subsets of D such that {o?crin} *s Cauchy in VF1;P(D). By assumption £ln C D, Cf7n D C.D,
and the Cauchy sequence converges to some / G W0'P(D}. By Theorem 2.4 (ii), C%(D) is compact in C(D] and there exist a subsequence, still denoted {c^n}, and an open set 0 C D such that
and hence in L p (D)-strong, since D is bounded. By uniqueness of the limit in L P (D), / = d£n and the Cauchy sequence d^n converges to d^ in W0'P(D). The other part of the proof is similar to that of part (ii). We have the following general result. Theorem 4.2. Let D be a bounded open domain in R N . (i) // {o?^n} weakly converges in Wl'p(D) for some p, I < p < oo, then it weakly converges in Wl'p(D] for all p, 1 < p < oo. (ii) If {dAn} converges in C(D], then it weakly converges in Wl'p(D) for all p, I < p < oo. Conversely if {dAn} weakly converges in Wl'p(D) for some p, I < p < oo, it converges in C(D}. (iii) Cd(D) is compact in W1>p(D)-weak for all p, 1 < p < oo.2 (iv) Parts (i)-(iii) also apply to C^(D}. Proof, (i) Recall that for D bounded there exists a constant c > 0 such that for all dA e Cd(D]
If {dAn} weakly converges in W /1 ' P (D), then
By Lemma 2.1 (iii) in Chapter 3 both sequences weakly converge for all p > 1, and hence {dAn} weakly converges in W1'P(D) for all p > 1. 2 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E', any closed ball is a compact metrizable space for the weak topology. Since Cd(D) is a bounded subset of the normed reflexive separable Banach space Wl>p(D), 1 < p < oo, the weak compactness of C^(D] coincides with the weak sequential compactness (cf. Dieudonne [1, II, Chap. XII, section 12.15.9, p. 75]).
176
Chapter 4. Topologies Generated by Distance Functions
(ii) If {d,An} converges in C(D), then by Theorem 2.2 (i) there exists dA e Cb(D) such that dAn -> oU in C(D) and hence in LP(D). So for all p e V(D}N,
By density of £>(£>) in L 2 (D), VeUn -»• VeU in L2(D)^-weak and hence dAn -»• cU in W1)2(D)-weak. From part (i) it converges in W1>p(.D)-weak for all p, 1 < p < oo. Conversely, the weakly convergent sequence converges to some / in W 1>P (Z)). By compactness of Cb(D) there exist a subsequence, still indexed by n, and dA such that d,An —>• rfyi in C(D] and hence in W1>p(D)-weak. By uniqueness of the limit, dA = f • Therefore, all convergent subsequences in C(D) converge to the same limit, so the whole sequence converges in C(D). This concludes the proof. (iii) Consider an arbitrary sequence {d,An} in Cd(D}. From Theorem 2.2 (ii) Cd(D] is compact and there exists a subsequence {d,An } and d,A & Cd(D) such that d,An —> dA in C(D). From part (ii) the subsequence weakly converges in W1>P(D) and hence Cd(D] is compact in W lip (Z>)-weak. Theorem 4.3. Let D be a closed domain in RN and {An} a sequence of nonempty sets converging to a nonempty set A of D in the Hausdorff topology:
Then
and for all compact subsets K of D
Corollary 1. Let D ^ 0 be a bounded open subset of R N and {Qn}, Qn ^ 0, a sequence of open subsets of D converging to an open subset Q, Q =£ 0, of D in the Hausdorff complementary topology:
Then
and
4. W^'P-Topology and Characteristic Functions
1 77
Proof of Theorem 4.3. For all x ^ A, eU(#) > 0 and
So for all £ ^ A
Finally, for all x G A and all n > 1
and the result follows trivially by taking the limsup of each term. We conclude that limsup XT < XA, n—>oo
and by using the analogue of Fatou's lemma for the limsup we get for all compact subsets K of D
In order to get the Lp-convergence of the characteristic functions of the closure of the sets in the sequence, we need the Lp-convergence of the gradients of the distance functions which are related to the characteristic functions of the closure of the sets (cf. Theorem 3.2 (iv)). We have seen in Example 4.1 that the weak convergence of the characteristic functions is not sufficient to obtain the strong convergence of the sequence {cUn } to d,A in Wl'2(D). However, if we assume that (x^ } is strongly convergent, it converges to the characteristic function XB of some measurable subset B of D. Is this sufficient to conclude that XB — XA^ The answer is negative. The counterexample is provided by Example 5.2 in Chapter 3, where
for some B C D such that
Remark 4.1. In view of part (ii) of Theorem 4.1 an optimization problem with respect to the characteristic functions xn of open sets Q in D for which we have the continuity with respect to xn can be transformed into an optimization problem with respect to d^ in W1'l(D] since For instance, this would apply to the transmission problem (3.3)-(3.6) in section 3.1 of Chapter 3.
Chapter 4. Topologiews Generated by Distance Functions
a
5
Sets of Bounded and Locally Bounded Curvature
Going from the uniform Hausdorff metric topology to the Wl'p'-topology readily extends the applicability of the distance function to problems involving the characteristic function. However, even for a bounded open hold-all D, the families Cd(D) and Cj(D) are closed but not compact in W1'p(D)-stTong (cf. Example 4.1). The situation is similar to that encountered for the set of characteristic functions X(.D) in the I/p(D)-topology, where we have introduced the Caccioppoli sets. Their natural analogue in Cd(D] and CJ(-D) are the sets introduced by Delfour and Zolesio [17, 32], which include Ck domains, convex sets, and Federer's [2] sets of positive reach. They lead to compactness theorems for Cd(D] and C%(D) in Wljp(D).
5.1
Definitions and Properties
Definition 5.1. (i) Given a bounded open set D in R N , a subset O of D, fJ ^ 0 (resp., Ci7 ^ 0), is said to be of bounded exterior (resp., interior] curvature with respect to Dif
Denote those families as follows:
(ii) A subset O of R N , $ 7 ^ 0 (resp., Cf2 ^ 0), is said to be of locally bounded exterior (resp., interior) curvature if
where B(x,p) is the open ball of radius p > 0 in x. From Theorem 5.1 and Definition 5.1 in Chapter 3, a function belongs to BVi oc (R N ) if and only if for each x e RN it belongs to BV(B(x, p)) for some p > 0. As in Theorem 5.2 of Chapter 3 for Caccioppoli sets, it is sufficient to satisfy this condition for points of the boundary d£l. Thus property (5.2) only depends on the properties of the boundary. Theorem 5.1. Let ft, 0 ^ 0 (resp., C$1 ^ 0)), be a subset o/R N . Then Q is of locally bounded exterior (resp., interior) curvature if and only ifVda (resp., VC/QQ) belongs to BVloc(RN)N. This theorem will be proved later as part (iii) of Theorem 5.3. Since we simultaneously deal with sets O and their complement Cf2, we shall often use the notation A to cover both cases. Recall that the relaxation of the perimeter of a set was obtained from the norm of the gradient of its characteristic
5. Sets of Bounded and Locally Bounded Curvature
179
function for Caccioppoli sets. For distance functions X~A = 1 ~~ \^^A and the gradient of d& also has a jump discontinuity along the boundary dA of magnitude at most 1. So it is not too surprising to discover that the closure of a set A with bounded interior or exterior curvature is a Caccioppoli set. Theorem 5.2. (i) Let D be a bounded open Lipschitzian subset o/R N . For any 0 C D, 0 ^ 0 (resp., 0 7^ CQ) such that
0 (resp., CO) has finite perimeter; that is,
(ii) For any subset Q o/R N ; 0 ^ 0 (resp., 0 ^ CO) swc/i t/iat
D (resp., Cf2) /ias locally finite perimeter; that is,
Proof. Given Vd^ in BV(D) 7V , there exists a sequence {uk} in C°°(D}N such that
as /c goes to infinity, and since |VcJ^(o;)| < 1, this sequence can be chosen in such a way that This follows from the use of mollifiers (cf. Giusti [1, Thm. 1.17, p. 15]). For all V in T)(D}N
For each Uk
where T Duk is the transpose of the Jacobian matrix Duk and
180
Chapter 4. Topologies Generated by Distance Functions
Figure 4.4. Vd^ for Examples 5.1, 5.2, and 5.3. since for W^'^Z^-functions ||V/||^i(£))N = \\^7f\\Mi(D')N• infinity
Therefore, as k goes to
where D2d,A is the Hessian matrix of second-order partial derivatives of d&- Therefore, Vxx e Ml(D}N. 5.2
Examples
It is useful to consider the following three simple illustrative examples (cf. Figure 4.4). Example 5.1 (half-plane in R2). Consider the domain
It is readily seen that
Thus Ad A is the one-dimensional Hausdorff measure of dA. Example 5.2 (ball of radius R > 0 in R 2 ). Consider the domain
5. Sets of Bounded and Locally Bounded Curvature
181
Clearly,
where HI is the one-dimensional Hausdorff measure. We see that Ad^ contains the one-dimensional Hausdorff measure of the boundary dA plus a term that corresponds to the volume integral of the mean curvature over the level sets of dA in CA Example 5.3 (unit square in R 2 ). Consider the domain
Since A is symmetrical with respect to both axes, it is sufficient to specify dA in the first quadrant. We use the notation Qi, Q2, Qa, and Q^ for the four quadrants in the counterclockwise order and c\, 02, 03, and 04 for the four corners of the square in the same order. We also divide the plane into three regions:
Hence
182
Chapter 4. Topologies Generated by Distance Functions
0, the h-tubular neighborhood of A is defined as
and the closed h-tubular neighborhood of A as
(ii) A function / : U —>• R defined in a convex subset U of RN is convex if for all x and y in U and all A e [0,1],
It is concave if the function — / i s convex. A function / : U —•> R defined in a convex subset U of RN is semiconvex (resp., semiconcave] if
is convex in U.
5. Sets of Bounded and Locally Bounded Curvature
183
(iii) A function / : U —> R defined in a subset U of RN is locally convex (resp., locally concave) if it is convex (resp., concave) in every convex subset of U. A function / : U —>• R defined in a subset U of RN is locally semiconvex (resp., locally semiconcave) if it is semiconvex (resp., semiconcave) in every convex subset of U. When U is convex the local definitions coincide with the global ones. In general, Uh(A) C Ah, but the equality does not necessarily hold. Lemma 5.1. Given a subset A, A ^ 0, o/R N and h > 0, the function
is convex in R N . In particular,
are, respectively, locally convex in RN \Ah and Uh(A). Proof. For all p € A define the convex function
Since tp is nonnegative, $L is convex and
By subtracting the constant term p|2 + h2 and the linear term —2 x • p from ^, we get the new convex function
Then the function
is finite for each x e R N , convex in x, and
184
Chapter 4. Topologies Generated by Distance Functions
If x is such that dA(x) > h, then for all p 6 A, \x — p\ > dA(x) > h and fc(a;) = \x\2 — 2hdA(x). If d,A(x] < /i, then there exists p £ A such that # — p\ < h and
Then, either for all p G A, \x — p\ < h and k(x) = \x\2 — h2 — d\(x) or, else, there exists p 6 A such that \x — p\ > /i,
and k(x) = \x 2 — d2A(x)~h2. We recover the result of Fu [1, Prop. 1.2] by observing that the restriction of k to RN \^4^ is locally convex. In addition, the function x 2 — d2A(x) is locally convex in Uh(A). This lemma has far-reaching consequences. Theorem 5.3. Let A be a nonempty subset o/R N . (i) The function fA(x) = \ (\x\2-d2A(x)) is convex mR N andVd\ €. BVioc(RN)N For all x and y in R N ,
or equivalently,
The set of projections T[A ( x ) onto A is a singleton PA (x) if and only if the gradient of fA exists. In that case
For all x and y in RN
The function d2A (x) is the difference
of two convex functions
(ii) VdA e BVioc(RN \A)N. More precisely, for all x 6 RN \A, there exists p > 0, 0 < 3p < dA(x], such that k&p is convex in B(x,1p) and hence VdA£BV(B(x,p))N.
6. Characterization of Convex Sets
185
(iii) A subset A ^ 0 of RN is of locally bounded exterior curvature if and only if VdA 0, \x\2 — d2A is locally convex in Uh(A) and hence in RN = (Jh>QUh(A). Therefore, \x\2 — d\ is convex in R N , and hence from Evans and Gariepy [1, Thm. 3, p. 240; Thm. 2, p. 239; and Aleksandrov's Theorem, p. 242], Vd2A G BVio C (R N ) ]V . Inequality (5.3) follows directly from the inequality Vx and y e R N , Vp(x) € KU(x),
0 such that VdA G ~BV(B(x, p ) } N . This is true in hit A, where Vd,A = 0, and in dA by assumption. It is also true for any point x in RN \A by part (ii). Therefore, Vc^ belongs to BVi oc (R N ) Ar and A is of locally bounded exterior curvature. Similarly, for an open subset Q of RN we only need to prove that for each x there exists p > 0 such that Vd Cfi e BV(B(x,p)}N. This is true in int Cfi, where Vd Cn = 0, and in dft by assumption. It is also true for any point x in RN \CO by part (ii). Therefore, VC^QQ belongs to BVioc(RN)'/v and £7 is of locally bounded interior curvature.
6
Characterization of Convex Sets
In the convex case the squared distance function is differentiable everywhere and this property can be used to characterize the convexity of a set. Theorem 6.1. Let A be a nonempty subset o/R N with convex closure. Then
186
Chapter 4. Topologies Generated by Distance Functions
(i) for each x G R N , HA(X)
=
{PA(X}} is a singleton and
(ii) d\ belongs to C^(RN) (and a fortiori to W^'C°°(RN)). Proof, (i) By definition
and since A is convex and z i—> \z — x\2 is strictly convex, there exists a unique PA(X] in A such that
(cf., for instance, Zarantonello [1, pp. 237-246] or Aubin [2, p. 24, Example 4.3]). So, for any two points x and y,
By adding up the above two inequalities,
(ii) From part (i) the map x i—> PA(X) is Lipschitz continuous and hence the map x i—>• Vd\(x) = 2[x— PA(X}} is also Lipschitz continuous. Therefore, o9A belongs toC^CR").
Theorem 6.2. (i) Let A be a nonempty subset of R N . Then (a) A convex =>• dyi convex; (b) rf^i convex =$> A convex; (c) Va; G R N , n^(a:) zs a singleton ^ A is convex.3 (ii) Let D be a nonempty open subset of R N . The subfamily
of Cd(D] is closed in C\OC(D}. It is compact in C(D] when D is bounded. The subfamily
of C%(D) is closed in C\OC(D}. It is compact in C(D] when D is bounded. 3
This part of the theorem is related to deeper results on the convexity of Chebyshev sets in metric spaces. A subset A of a metric space X is called a Chebyshev set provided that every point x of X has a unique projection PA (x) in A. The reader is referred to Klee [1] for details and background material.
6. Characterization of Convex Sets
187
(iii) For all subsets A of RN such that dA 7^ 0 and A is convex, Vd^ belongs to BVioc(R N )^> and the Hessian matrix D2d^ of second-order derivatives is a matrix of signed Radon measures that are nonnegative on the diagonal. Moreover, dA has a second-order derivative almost everywhere, and for almost all x and y in R N ,
as y —* x. Proof, (i) (a) Given x and y in R N , there exist x and y in A such that cU(#) = \x—x\ and cU(y) = \y~y\- By convexity of A, A is convex, and for all A, 0 < A < 1, A x + (I - X)y G A and
and dA is convex in R N . (b) If dA is convex, then
But x and y in A imply that d^ in C(D}. If £7 is empty, there is nothing to prove. If fi is not empty, consider two points x and y in £7 and A £ [0,1]. There exists r > 0 such that
There exists N > Q such that for all n > N, \\d^n — A can be convex or semiconvex. However, it is never semiconcave in RN for nontrivial sets. Theorem 6.4. (i) Given a subset A, A ^ 0 o/R N ,
if and only if is locally convex in Uh(A). (ii) Given a subset A o/R N such that 0 ^ A ^ R N ,
Proof, (i) If fc is convex in R N , it is locally convex in Uh(A). Conversely, assume that there exists h > 0 and c > 0 such that fc is locally convex in Uh(A). From Lemma 5.1,
is locally convex in RN \Afl/2- Therefore, for c = max{c, 1/h}, the function fc is locally convex in RN and hence convex on R N . (ii) Assume the existence of a c > 0 for which fc is convex in RN. For each y 6 A the function
is also convex since it differs from fc(x) by the linear term
6. Characterization of Convex Sets
191
Since 0 / A =£ R N , there exist x E CA and p <E A such that
For any t > 0, define x t = p — t (x — p] and A = t/(l + t) G ]0,1[ and observe that
But
since by construction C[A(X} > 0 and cd^(x) < 1/2. Therefore for t, 0 < t < 1, the above quantity is strictly negative, and we have constructed two points x and Xt and a A, 0 < A < 1, such that
This contradicts the convexity of the function x i-+ Fc(x,p) and a fortiori of fc. Theorem 6.5. Let A be a nonempty subset o/R N for which the following condition is satisfied: Vx G dA, 3/9 > 0 such that d^ is semiconvex in B(x,p). Then the gradient of dA belongs to BVi oc (R' N ) 7V Proof. For all x e int A, dA = 0 and there exists p > 0 such that B(x,p) C int^4 and the result is trivial. For each x € R N \A, dA(x) > 0. Pick h = d^(x)/2. Then for all y G B(x, h) and all a G A,
and B(x,h) C RN \Ah- By Lemma 5.1 the function \x 2 /(2/i) — ^A^) is locally convex in RN \A^ and a fortiori convex in B(x,h). Finally, by assumption for all x E 0 such that dA is semiconvex. Hence for each x G R N , VcU belongs to BV(5(x,p/2)) for some p > 0. Therefore, VdA belongs to BV loc (R N ) N .
192
7
Chapter 4. Topologies Generated by Distance Functions
Federer's Sets of Positive Reach
For closed submanifolds of codimension larger than 1, £1 — 0 such that HA(X) = {PA(X}} is a singleton for every x G Uh(A). The maximum h for which the property holds is called the reach of A and denoted reach (A). For a convex set A, reach (A) — +00. All nonempty convex sets have positive reach. Bounded domains and submanifolds of class C2 also have positive reach. A quite impressive result of Federer was to make sense of the classical SteinerMinkowski formula for that class of sets. He also showed that for all r, 0 < r < h, the boundaries of the closed tubular neighborhoods Ar are C1'1-submanifolds of codimension 1 in R N . The next theorem summarizes several characterizations of sets with positive reach. Note that condition (vii) is a global condition on the smoothness of d2A in the tubular neighborhood Uh (A) that is similar to the one of Definition 5.1(ii) on Vefo in a neighborhood of 50. Theorem 7.1. Given a nonempty subset A of R N , the following conditions are equivalent. (i) 3/i > 0 such that d& belongs to C^(Uh(A)\A). (ii) 3/i > 0 such that dA belongs to Cl(Uh(A}\A). (iii) 3h > 0 such that MX e Uh(A)\A, HA(X) is a singleton. (iv) 3h > 0 such that Vx € Uh (A), HA (x) is a singleton. (v) A has positive reach, that is, reach (A) > 0. (vi) 3/i > 0 such that PA belongs to C^(Uh(A)). (vii) 3/i > 0 such that d2A belongs to C^(Uh(A)). Proof. The elements of the proof can be found in Federer [1], (i) =>• (ii) =^> (iii) =$• (iv) => (v) are obvious. (v) =$>• (vi) For each x e Uh(A), HA(X) = {p(x)} is a singleton, and for all t > 0 such that £d^(rr) < h,
By assumption for all t > 0, td,A(x] < h, and the projection of p(x) + t(x — p(x}} onto A is unique and equal to p(x). Otherwise we could reduce d^(x), leading to a contradiction. Hence for all t > 0, tdA(x) < h,
7. Federer's Sets of Positive Reach
193
For all a G A, t/ G Uh(A) and t > 0 such that £cU(y) < h
So for any yi and y2 in RN and t > 0 such that tcUG/i) < ^ an there exists p > 0 such that cU (#)+/» < /i. Let t = /i/(cU(:r)+p), which is strictly greater than 1. For all y £ B(x,p), d,A(y) < AA(X} + p and t d A ( y ) < h. Therefore, for all y\ and y^ in B(x,p]
The result follows from the fact that any compact subset K of Uh (A] can be covered by a finite number of neighborhoods B(xi, p i ) , Xi G K, pi > 0. (vi) => (vii) The proof follows from the identity PA(X) = x — ^Vd2A(x). (vii) =>• (i) For each x e Uh(A)\A, there exists p, 0 < /o, such that dA(x}+p < h. Therefore, for all y G 5(x,p), /i > ^A(^) + /o > cU(y) > ^A(^) — P > 0, and d^ G Cl'l(B(x,p}}. For any yi and y2 in B(x,p),
So, from the proof of (v) =4> (vi) and d A (y 2 ) > d^(x) — p > 0,
and Vd A £ C10'1 (•£?(£)/?))> d A G C 1)1 (5(x,p)). Therefore, the property holds in any compact subset of Uh(A)\A and dA G C110'c1(Vh(A)\A).
194
8
Chapter 4. Topologies Generated by Distance Functions
Compactness Theorems
Subsets of a bounded hold-all D with a positive reach greater than or equal to some h > 0 form a compact family of sets (cf. Federer [1, Thm. 4.13]). Theorem 8.1. Let D be a fixed bounded open subset o/R N . Let {An}, An ^ 0, be a sequence of subsets of D. Assume that there exists h > 0 such that
Then there exist a subsequence {Ank} and A C D, A ^ 0, such that d\ £ C&(Uh(AD and
This is a compactness theorem similar to the compactness of Cd(D) in C(D] for a bounded open subset of R N . For the family of sets with bounded interior or exterior curvature, the key result is the compactness of the embeddings
for bounded open Lipschitzian subsets D of RN and p, 1 < p < oo. It is the analogue of the compactness theorem (Theorem 5.3 of Chapter 3) for Caccioppoli sets
which is a consequence of the compactness of the embedding
for bounded open Lipschitzian subsets D of RN (cf. Morrey [1, Def. 3.4.1, p. 72; Thm. 3.4.4, p. 75] and Evans and Gariepy [1, Thm. 4, p. 176]). As for characteristic functions in Chapter 3, we give a first version involving global conditions on a fixed bounded open Lipschitzian hold-all D. In the second version the sets are contained in a bounded open hold-all D with local conditions in their tubular neighborhood or the tubular neighborhood of their boundary. 8.1
Global Conditions on D
Theorem 8.2. Let D be a nonempty bounded open Lipschitzian subset o/R N . The embedding (8.2) is compact. Thus for any sequence {&,„.}, ® ^ Qn, of subsets of D such that
8. Compactness Theorems
195
there exist a subsequence {Qnk} and a set £1, 0 ^ 0, such that Vcfo G BV(D) Af and for all
andxn GBV(£>). Proof. Given c > 0 consider the set
By compactness of the embedding (8.5), given any sequence {cfo^}, there exist a subsequence, still denoted {dnT,}, and / G EV(D}N such that Vcfon —» / in Ll(D}N. But by Theorem 2.2 (ii), C d (D) is compact in C(D) for bounded D and there exist another subsequence {dnn } and d$i G Cd(D] such that d^r, "^ ^n m C(D) and, a fortiori, in Ll(D). Therefore, dnn converges in W1'1(D) and also in Ll(D}. By uniqueness of the limit / = Vefo and d^7i converges in Wl'l(D) to cfo. For $ G Vl(D}NxN as fc goes to infinity
D2d$i \MI(D) ^ c? and Vd^ G BV(Z)) 7V . This proves the compactness of the embedding for p = I and properties (8.7). The conclusions remain true for p > I by the equivalence of the VF^-topologies on Cd(D] in Theorem 4.1(i). When D is bounded open, C^(D) is compact in C(D] and closed in W lip (D), 1 < p < oo, and we have the analogue of the previous two compactness theorems. Theorem 8.3. Let D be a nonempty bounded open Lipschitzian subset o/R N . The embedding (8.3) is compact. Thus for any sequence {£ln} of open subsets of D such that there exist a subsequence {Qnk} ana an open subset £1 of D such that V^cn £ BV(D}N and
for all p, I < p < oo.
ana?^(EBV(L>).
196
Chapter 4. Topologies Generated by Distance Functions
Proof. Given c> 0 consider the set
By compactness of the embedding (8.5), given any sequence {d^n} there exist a subsequence, still denoted {^cnn}? an / in L1(D)N. But by Theorem 2.4 (if), C%(D) is compact in CQ(D) for bounded D and there exist another subsequence { / in Ll(UB)N. Since Uh(D) is bounded, Cd(Uh(D}} is compact in C(Uh(D}} and there exists another subsequence, still denoted {d^n}, and 0 ^ 0 C D such that efon —» d^ in C(Uh(D)} and, a fortiori, in Ll(Uh(D}}. Therefore, d$in converges in WI>I(UB] and also in LI(UB)- By uniqueness of the limit, / = Vc?n on UB and d^,, converges to cfo in
198
Chapter 4. Topologies Generated by Distance Functions
Wl'l(UB}- By Definition 5.2 and Theorem 5.3 (iii), Vcfo and Vcfon all belong to BVi oc (R N ) JV since they are BV in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), %^ E BVi oc (R N )- The above conclusions also hold for the subset t/j l _2 £ (fi) of UB(ii) Convergence in W1'p(Ufl(D)). Consider the integral
Prom part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of Ct// l _2e(fi). From (8.16) for all n>N,
The second integral reduces to
which converges to zero since Vcfon —*• Vcfo in L2 (Uh(D))N-weak in part (i) and the fact that |Vcfo| = 1 almost everywhere in Uh(D)\Uh-2e(ty- Therefore, since dfln-^dnmC(U^(D)),
and by Theorem 4.1(i) the convergence is true in W1>p(C7/l(Z))) for all p > 1. (iii) Properties (8.14). Consider the initial subsequence {dnn} which converges to d$i in H1 (Uh(D))-weak constructed at the beginning of part (i). This sequence is independent of e and the subsequent constructions of other subsequences. By convergence of dnn to cfo in Hl(Uh(D))-weak for each $ e T>l(Uh(fy)NxN,
Each such $ has compact support in t/i^fi), and there exists e ~ £•($) > 0, 0 < 3e < /i, such that
From part (ii) there exists N(e) > 0 such that
8. Compactness Theorems
199
For n > N(e) consider the integral
By convergence of Vl(Uh(tt))NxN
Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof. Proof of Lemma 8.1. First check that
It is sufficient to show that all points x of int 0 are contained in the right-hand side of the above expression. If ddn(x) < h, then x G Uh(dft); if ddti(x) > /i, then B(x,ti) C 0 and x E ft-h. For all $ e T>l(Uh(ty}NxN, $ has compact support and there exists £ > 0 , 0 < 2 e < / i , such that
But Uh-e(ty C Uh(Q) and there exists {xj G fJ_h : 1 < j < m} such that
Let i/>0 G P(C/h.(5ri)) and ^ G T>(B(xj,h}} be a partition of unity such that
Consider the integral
200
Chapter 4. Topologies Generated by Distance Functions
By construction for ||$||£7(1^(0)) < 1
But for j, 1 < j < m, B(xj,h] C intfJ, where both d^ and Vcfo are identically zero. As a result the above integral reduces to
and this completes the proof. Theorem 8.5. Let D be a nonempty bounded open subset o/R N . Let {fJn} be a sequence of nonempty open subsets of D and assume that there exist h > 0 and c > 0 such that
Then there exist a subsequence {Slnk} and an open subset J7 of D such that VC?QQ G BVi oc (R N ) 7V , and for all p, l° (£//»((&)),
and xsi belongs to BVi oc (R N )Proof. First note that 0 and let 17 be an open subset of a nonempty bounded open subset D o/R N . Then
Ifdtt^0,
8. Compactness Theorems
201
The proof of this lemma will be given after the proof of the theorem. (i) By Theorem 2.4 (ii) since D is bounded, there exists a subsequence, still denoted {dcnn}> and an open subset f2 of D such that
and another subsequence, still denoted {dfmn}, such that
For all e > 0, 0 < 3e < h, there exists TV > 0 such that for all n > N
Clearly, since fi C D and On C D and D is open, C^0n ^ 0 and C^O / 0. Furthermore,
From (8.17) and (8.20),
As in part (i) of the proof of Theorem 8.4, we can construct a bounded open Lipschitzian set UB between Uh-2e(^rjft) and [/^-^(C^O) such that
Since UB is bounded and Lipschitzian, it now follows by compactness of the embedding (8.5) for UB that there exists a subsequence, still denoted {d(jQn}, and / € BV(UB)N such that Wc^ -> / in Ll(UB}N. Since L> U Z7B is bounded, C^(D U £/#) is compact in C$(D U C/B), and there exists another subsequence, still denoted {d^n}, and an open subset Q of D such that d^n —>• ^Q^ in Co(D U t/s) and, a fortiori, in L^DUt/e). Therefore d^n converges in WI>I(UB) and also in LI(UB}- By uniqueness of the limit, / = Vdco on BD and c/Q Qj converges to d^Q in H /I ' I ([/B). By Definition 5.2 and Theorem 5.3 (iii), V^QQ and V^QQ^ all belong to BVioc(R'N)JV since they are BV in tubular neighborhoods f/^(C^Jl) and Uh(£>pQn} of their respective boundaries 90 and d£ln. Moreover, by Theorem 5.2 (ii), xcn ^ BVi oc (R N ) and, a fortiori, xn since 17 is open. The above conclusions also hold for the subset Uh-2e(^>D^ °^ UB(ii) Convergence in WQ'P(D}. Consider the integral
202
Chapter 4. Topologies Generated by Distance Functions
From part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of C^f//1_2e(C^O). From the relations (8.21) for all n > N,
The second integral reduces to
which converges to zero by weak convergence of Wcfin to Vo^ in L2(D)N and the fact that |Vdg^| = 1 almost everywhere in J D\C// l _2 £ (C^fi). Therefore, since d Cttn ->• dCfi in C0(D) and by Theorem 4.1 (i) the convergence is true in WQ'P(D) for all p > 1. (iii) Properties (8.19). Consider the initial subsequence {d^n}, which converges to rffjo in HQ(D U [/h(Cfi))-weak constructed at the beginning of part (i). This sequence is independent of e and the subsequent constructions of other subsequences. By convergence of d^ to d^ in H$(D U ?7^(CO))-weak for each
fcePH^Cn))^",
Now each $ G P 1 (C// l (Cfi))^ Vx7V has compact support in [/^(CO) and there exists e = e($) > 0, 0 < 3e < h, such that From part (ii) there exists N(e) > 0 such that
For n > N(e) consider the integral
By convergence of VdCfln to WCn in L2(JDUC//l(CO))-weak, for all $ e T>l(Uh(£ty)N>
Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.
8. Compactness Theorems Proof of Lemma 8.2. The proof is similar to that of Lemma 8.1 with
since both d^ and VC^CQ are identically zero on CD.
203
This page intentionally left blank
Chapter 5
priented Distance Function and Smoothness of Sets
1
Introduction
In this chapter we study oriented distance functions and their role in the description of the geometric properties of domains and their boundaries. They are also known as algebraic or signed distance functions. Our choice of terminology emphasizes the fact that, for a smooth domain, the associated oriented distance function defines an orientation of the normal to the boundary. They enjoy many interesting properties. For instance, they retain the nice properties of the distance functions but also generate the classical geometric properties associated with sets and their boundaries. The smoothness of the oriented boundary functions in a neighborhood of the boundary of the set is equivalent to the smoothness of its boundary. Similarly, the convexity of the function is equivalent to the convexity of the closure of the set. In addition, their respective gradient and Hessian matrix, respectively, coincide with the unit outward normal and the second fundamental form on the boundary of the set. Finally, they provide a framework for the classification of domains and sets according to their degree of smoothness, much like Sobolev spaces and spaces of continuous and Holderian functions do for functions. The first part of the chapter deals with basic definitions, constructions, and results. The second part specializes to specific subfamilies of oriented distance functions. The last part concentrates on compact families of subsets of oriented distance functions. In the first part, section 2 presents the basic properties, introduces the uniform metric topology, and shows its connection with the Hausdorff and complementary Hausdorff topologies of Chapter 4. Section 3 is devoted to the differentiability properties, the associated set of projections onto the boundary, and completes the treatment of skeletons and cracks. Section 4 gives the equivalence of the smoothness of a set and the smoothness of its oriented distance function in a neighborhood of its boundary for sets of class C1'1 or better. When the domain is sufficiently smooth the trace of the Hessian matrix of second-order partial derivatives on the boundary is the classical second fundamental form of geometry. Section 5 deals with the VF1'p-topology on the set of oriented distance functions 205
206
Chapter 5. Oriented Distance Function and Smoothness of Sets
and the closed subfamily of sets for which the volume of the boundary is zero. In the second part of the chapter, in section 6, we study the subfamily of sets for which the gradient of the oriented distance function is a vector of functions of bounded variation: the sets with global or local bounded curvature. They are rather large classes for which quite general compactness theorems will be proved in section 9. Some examples are given to illustrate the behavior of the norms in tubular neighborhoods as the thickness of the neighborhood goes to zero in section 6.2. Section 6.3 introduces Sobolev or Ws'p-domains, which provide a framework for the classification of sets according to their degree of smoothness. For smooth sets this classification intertwines with the classical classification of Ck- and Holderian domains. Section 7 extends the characterization of closed convex sets by the distance function to the oriented distance function and introduces the notion of semiconvex sets. The set of equivalence classes of convex subsets of a compact hold-all D is again compact, as it was for all the topologies considered in Chapters 3 and 4. Section 8 shows that sets of positive reach introduced in Chapter 4 are sets of locally bounded curvature and that their boundary has zero volume. In the last part of the chapter, section 9 gives compactness theorems for sets of global and local bounded curvatures from a uniform bound in tubular neighborhoods of their boundary. They are the analogues of the theorems of Chapter 4. Finally, section 10 gives the compactness of the sets of Lipschitzian domains in a bounded hold-all under the uniform cone property for the VFlip-topology associated with the oriented distance functions. We recover as a corollary the compactness of Theorem 5.9 of section 5.4 in Chapter 3 for the associated family of characteristic functions in LP(D}. Furthermore, we get the compactness of the set of characteristic functions of the complements in LP(D] and the compactness of the associated families of distance functions and distance functions of the complement in W1A = 0. This condition is completely equivalent to dA ^ 0, in which case bA coincides with the algebraic distance function to the boundary of A:
Noting that b^A = —bA, it means that we have implicitly chosen the negative sign for the interior of A and the positive sign for the interior of its complement. We shall see later that for sets with a smooth boundary, the restriction of the gradient of bA to dA coincides with the outward unit normal to the boundary of A. Changing the sign of bA gives the inward orientation to the gradient and the normal. Theorem 2.1. Let A be a subset o/R N . Then (i) A ^ 0 and (L4 ^ 0 ~B = A and CU = £B dA — dA.
(iii) (iv) (v)
(vi) If dA ^ 0, the function bA is uniformly Lipschitz continuous in RN and
Moreover, bA is (Frechet) differentiable
almost everywhere and
208
Chapter 5. Oriented Distance Function and Smoothness of Sets
Proof, (i) The proof is obvious, (ii) By assumption,
Since A C D and B C D, then £A D Cl) and C# D C.D and
Also, since B C D for all x e B,
and a fortiori in D. The equality case follows from the fact that bA = 65 if and only if bA > &s and 6^ ddA(x). (iv) If 6.4 = d,A — d£A > 05 then rf^ > d$A and A C C^4, and necessarily A C dA and A = 0. (v) bA = 0 is equivalent to 6^ > 0 and b^A — ~bA > 0. Then we apply (v) twice. But CU = dA = A int C^ = 0 = int A dA = R N . (vi) Clearly,
2. Uniform Metric Topology
209
For x e A and y e int CA = CA, cU(j/) > 0 and
By assumption B(y,dA(y}) C intCA Define the point
The argument is similar for x e int A and i/ £ C A The differentiability follows from Theorem 2.1 (vii) in Chapter 4.
2.2
Uniform Metric Topology
From Theorem 2.1 (i) the function b^ is finite at each point when dA ^ 0. This excludes A = 0 and A = R N . The zero function &A(#) = 0 Vx G RN corresponds to the equivalence class of sets A such that
This class of sets is not empty. For instance, choose the subset of points of RN with rational coordinates or the set of all lines parallel to one of the coordinate axes with rational coordinates. Let D be a nonempty subset of RN and associate with each subset A of D, dA 7^ 0, the equivalence class
and the family of equivalence classes
The equivalence classes induced by 6^ are finer than those induced by G?A since both the closures and the boundaries of the respective sets must coincide. As in the case of dA we identify ^(D) with the family Cb(D] through the embedding
When D is bounded, the space C(D] endowed with the norm H/Hc^D) is a Banach space. Moreover, for each A C D, 6^ is bounded, uniformly continuous on D, and bA G C(D}. This will induce the following complete metric:
on Fb(D).
210
Chapter 5. Oriented Distance Function and Smoothness of Sets
When D is open but not necessarily bounded, we use the space C\OC(D) defined in section 2.2 of Chapter 4 endowed with the complete metric p$ defined in (2.8) for the family of seminorms {QK} defined in (2.7). It will be shown below that Cb(D] is a closed subset of C\OC(D] and that this induces the following complete metric on K(D):
The following subfamilies:
of Fb(D) and Cb(D) will be important. We now have the equivalent of Theorem 2.2 of Chapter 4. Theorem 2.2. Let D, 0 ^ D, be an open (resp., bounded open) subset o/R N . (i) The set Cb(D] is closed in C\OC(D] (resp., C(D)), and ps (resp., p] defines a complete metric topology on Fb(D}. (ii) For a bounded open subset D o/R N , the set Cb(D) is compact in C(D}. (iii) For a bounded open subset D o/R N , the map
is continuous: for all b^ and bs in Cb(D],
Proof, (i) It is sufficient to consider the case for which D is bounded. Consider a sequence {An} C D, dAn ^ 0 such that b&n —> / in C(D) for some / in C(D). Associate with each g in C(D] its positive and negative parts
Then by continuity of this operation
By Theorem 2.2 (i) of Chapter 4, there exists a closed subset F, 0 ^ F C D, such that
2. Uniform Metric Topology
211
Moreover, F ^ RN since D is bounded. By Theorem 2.4 and the remark at the beginning of section 2.3 of Chapter 4, there exists an open subset G C -D, G ^ R N , such that
Therefore,
Define the sets
They form a partition of R N , RN = A~ U A° U A+, and
If A° was empty, R N could be partitioned into two nonempty disjoint closed subsets. Since A° is closed
Moreover, since A" and A+ are open, A" C A~ U A°, A+ C ^4+ U A°,
Let Q be the subset of points in RN with rational coordinates. Define
and notice that by density of Q and CQ in R N ,
212
Chapter 5. Oriented Distance Function and Smoothness of Sets
Consider the following new partition of RN:
Define
and
But
and since dA~ U 8A+ = dA°
(ii) For the compactness, consider any sequence {&An} C Cb(D] C C0)1(.D). Since D is bounded, bAn and V6,4n are both pointwise uniformly bounded in D. From Theorem 2.2 in Chapter 2 the injection of C°'l(D) into C(D] is compact and there exist / € C(D] and a subsequence {6^n } such that 6^n —>• / in C(D}. From the proof of the closure in part (i), there exists A C D, dA -•£ 0, such that / = &A(iii) For all 6,4 and bs in Cb(D) and rr in Z),
Moreover, dA = b\ — (\b&\ + ^)/2 and C?QA = b~A = (|6^| — &A)/2, and necessarily
By combining the above three inequalities we get (2.8). We have the analogue of Theorem 2.3 of Chapter 4 and its corollary for A, CA and dA. In the last case it takes the following form (to be compared with Richardson [1, Lem. 3.2, p. 44] and Kulkarni, Mitter, and Richardson [1] for an application to image segmentation):
3. Projection, Skeleton, and Differentiability of bA
213
Corollary \. Let D be a nonempty open (resp., bounded open) subset o/R N . Define for a subset S of RN the sets
(i) Let S be a subset o/R N . Then Hb(S) is closed in C\OC(D) (resp., C(D)). (ii) Let S be a dosed subset o/R N . Then Ib(S) is closed in C\OC(D) (resp., C(D}}. If, in addition, SC\D is compact, then Jb(S) is closed in C\OC(D) (resp., C(D}}. (\\\) Let S be an open subset o/R N . Then Jb(S) is open in C\OC(D) (resp., C(D)). If, in addition, CSTl-D is compact, then Ib(S) is open in C\OC(D) (resp., C(D)). (iv) For D bounded, associate with an equivalent class [A]b the number #b([A]b) = number of connected components of dA. Then the map is lower semicontinuous. Proof. This proof is the same proof as for Theorem 2.3 of Chapter 4 using the Lipschitz continuity of the map b^ |~^ dgA — \^A from C(D) to C(D).
3
Projection, Skeleton, and Differentiability of 6^
In this section we study the connection between the gradient of 6,4 and the projection onto dA and the characteristic functions associated with dA. We further relate the set of singularities of the gradients and the notions of skeleton1 and set of cracks. Definition 3.1.
Let A be a subset of RN such that 0 ^ dA. (i) The set of projections of x onto dA is given by
The elements of HQA(X] are called projections onto dA and denoted by PQA(X).). (ii) The set of points where the projection onto dA is not unique,
is called the skeleton of A. Since TlgA(x) is a singleton for x 6 dA, then 1
Our definition of a skeleton does not exactly coincide with the one used in morphological mathematics (cf., for instance, Matheron [1] or Riviere [1]).
214
Chapter 5. Oriented Distance Function and Smoothness of Sets
(iii) The set of cracks is defined as
The set of singularities, Sing(V6^), of V&A has zero ^'-dimensional Lebesgue measures since 6^ is Lipschitz continuous and hence differentiable almost everywhere. The function b^ enjoys properties similar to the ones of &A and d§A since (6^4 = ddA and we have the analogue of Theorem 3.1 in Chapter 4. Theorem 3.1. Let Abe a subset o/R N such that 0 ^ dA, and let x 6 R N . Define
(i) The set HQA (x) is nonempty, compact, and
(ii) For all x and v in RN
where &B is the support function of the set B,
and coB is the convex hull of B. (iii) The following statements are equivalent: (a) b\(x) is (Frechet) differentiable atx, (b) b\(x) is Gateaux differentiable at x, (c) HgA.(x) is a singleton. Henceforthh
(iv) When b\ is differentiable
at x, 110,4(a;) = {PQA(X}} is a singleton, and
3. Projection, Skeleton, and Differentiability of 6^
For all x G dA, H.QA(X] = {x}, b\ is differentiable Hence
215
at x and Vb2A(x] = 0.
Proof. The result follows from Theorem 3.1 of Chapter 4 using the identity b2A = dlA,
Remark 3.1. The uniqueness of the projection pdA(%) at x is equivalent to the existence of V6^(x). In both cases the identity
is satisfied, and all the properties of PQA can be obtained from those of b2A or JQAWe now compute the gradient of bA and relate it to the characteristic functions of 0A Theorem 3.2. Let A be a subset o/R N such that 0 ^ dA. (i) // Vb^(x) exists at a point x in R N , then HQA(X) — {POA(X)} is a singleton,
In particular, for almost all x € dA, V6^(x) = Vdg^(x) = 0,
and the set
has zero N-dimensional Lebesgue measure. (ii) For all x G dA, V6^(:c) exists and is equal to 0. For all x £ QA, bA is differentiable at x if and only if b\ is differentiable at x. In particular,
and the last identity is satisfied almost everywhere in R N .
216
Chapter 5. Oriented Distance Function and Smoothness of Sets
(iii) Given x G R N , a G [0,1], p G T\.dA(x), and xa = p + a (x — p), then
In particular if HQA(X) is a singleton, then HdA(xa) is a singleton and Vb\(xa) exists for all a, 0 < a < 1. //, in addition, x ^ dA, then V6^(x a ) exists for allO x and b^ is differentiable at x. In the other direction the result is trivial. (iii) The proof of (iii) is straightforward. Remark 3.2. For sufficiently smooth domains, the subset d^A of the boundary dA coincides with the reduced boundary of finite perimeter sets. Remark 3.3. In general, VdA(x) and Vdrj A (x) do not exist for x € dA. This is readily seen by constructing the directional derivatives for the half-space
at the point (0,0). Nevertheless, V6^(0,0) exists and is equal to (1,0), which is the outward unit normal at (0,0) € dA to A. Note also that for all x G 2, and a real number i, 0 < i < 1, then for each x G dA, there exists a neighborhood W(x] of x such that 6^4 G Ck'\W(x}} and b\ G C^l(W(x)). Proof, (i) From the definition of a set of class C1'1 for each x G dA, there exists a bounded open neighborhood U(x) of x where the set A can be locally described by the level sets of the C1'^function
since by definition
Denote by cx the Lipschitz constant of V/ in U(x). The boundary dA is the zero level set of / and the gradient
is normal to that level set. Thus the outward normal to A on dA is given by
There exists a bounded open neighborhood V(x) of x, and a. > 0 such that V(x) C U(x),
By the characterization of dA for all y G V(x),
We know that the set of minimizers
220
Chapter 5. Oriented Distance Function and Smoothness of Sets
is nonempty and compact. By using the Lagrange multiplier theorem for the Lagrangian
the elements p of Hg^d/)
are
characterized by the existence of a X(p] G R such that
since V/(jp) ^ 0 on dA. The next question is the uniqueness of p. Construct the function
and show that, for y sufficiently close to dA, F(y,p) has a unique fixed point p. Consider for z2 and z\ in V(x) the difference
In particular, the outward unit normal is Lipschitzian
Coming back to (4.3),
Choose the new neighborhood of x:
So for all y € W(x)
4. Boundary Smoothness and Smoothness of 6^
221
is a contraction and there is a unique p(y) G V(x) such that
By construction of W(o;)
Since HdA(y) 7^ 0 and p(y] is unique, II^ (y) = [p(y)} is necessarily a singleton. By Theorem 4.1, 6^ e Cl'l(W(x)) and pa^ e C^V^)). By Theorem 3.1 (iii) for y e W(x)\dA, V&^y) exists and
where
Both N and p are Lipschitzian and hence the composition N o p is Lipschitzian. Moreover, since &A is differentiable almost everywhere and m(dA) — 0
and necessarily VbA 6 C0'1^^))^, &A E Cl>l(W(x)). In view of (4.2), V6A = n o p. This proves the theorem when A is of class C1'1. (ii) When A is of class C fe '^ for an integer fc, /c > 2, and a real number £, 0 < t < 1, it is C1'1, and the previous constructions and conclusions remain true. Consider the map
From the first part, there exists a unique p(y) G V(x) such that
where
By assumption, / is at least C2 in V(x) and hence bounded in some smaller neighborhood V'(x), V'(x) C V(x). By putting a smaller bound on |6>i(y)| there exists a smaller neighborhood W'(x) of x in W(x) such that DzG(y,z] is invertible for all (y,z) G W'(x) x V'(x). So the conditions of the implicit function theorem are
222
Chapter 5. Oriented Distance Function and Smoothness of Sets
met. There exists a neighborhood Y(x] C V'(x) of x and a unique Cl- (and hence Ck~lji-} mapping
(cf., for instance, Lang [1, Prop. 5.3, p. 15] and Schwartz [3, Thm. 31, p. 299]). In view of the fact that (4.4) is satisfied everywhere in Y(x),
then V6^ belongs to Ck~1'e(Y(x))N as the composition of two Ck~^^ maps. Therefore, bA is Ck^ in Y(x).
Example 4.1. Consider the two-dimensional domain
defined as the epigraph of the function
for some arbitrary integer n > 1. We claim that fJ is a set of class C1:1~1/fn and
In view of the presence of the absolute value the point (0,0) is the point where the smoothness of d£l will be minimum. Therefore, there exists no neighborhood of (0,0) where the derivative Vfo exists and &Q is not Cl and a fortiori not C1'1"1/™. Indeed, for each (0, y)
The point (0,7/), y > 0, belongs to Sk(il) if there exist two different points x that minimize the function
Since / is symmetric with respect to the y-axis it is sufficient to show that there exists a strictly positive minimizer x > 0. A locally minimizing point x > 0 must satisfy the conditions
Equation (4.6) can be rewritten
4. Boundary Smoothness and Smoothness of 6^
223
and x = 0 is a solution. The second factor can be written as an equation in the new variable X = xl/n and
It has exactly one solution X since
g(0) = — y and g(x) goes to infinity as X goes to infinity. In particular, X > 0 for y > 0 and X = 0 for y = 0. For y > 0 and x = Xn
and
Therefore, x is a local minimum. To complete the proof for y > 0 we must compare F(0, y} = y2 for the solution corresponding to x = 0 with
for the solution x > 0. Again using identity (4.8),
This proves that for y > 0, x > 0 is the minimizing positive solution. Therefore, for all n > 1, f(x] = \x 2 ~ 1 / n yields a domain fi for which the strictly positive part of
224
Chapter 5. Oriented Distance Function and Smoothness of Sets
the y-axis is the skeleton. It is interesting to note that, for each point x, the points (x, /(#)) G 1, / £ C1'1. For n = 1, / £ C0'1, and for n > 1, / € Cl'l~l/n. This gives an example in dimension 2 of a domain 0 of class )
225
Proof. For simplicity, we use the notation b = b^ and p — pr(i) The result follows from Theorem 4.3 (i). (ii) By assumption from part (i) the function b belongs to Cl>l(W(x)) and hence p e C°'1(W(x)\ R N ). Thus, almost everywhere in W(x),
Now
But since p and Tt belong to C0jl(W(x}]'RN}1 the identity necessarily holds everywhere in W(x). Moreover,
Finally
In particular, almost everywhere in W(x),
5
W1'p(D)-Topology and the Family C£(D)
From Theorem 2.1 (vi) bA is locally Lipschitzian and belongs to W^(R N ) for all p > 1. So the previous constructions for d& and d^A can be repeated with the space W\QC (R N ) m place of Cioc(-D) to generate new Wl'p metric topologies on the family Cb(D}. Moreover the other distance functions can be recovered from the map
and the characteristic functions from the maps
One of the advantages of the function 6^ is that the Wl^-convergence of sequences will imply the I/p-convergence of the corresponding characteristic functions of int A, int CA, and dA, that is, continuity of the volume of these sets. In this section we give the analogues of Theorems 4.1 and 4.2 in Chapter 4.
226
Chapter 5. Oriented Distance Function and Smoothness of Sets
Theorem 5.1. Let D be an open (resp., bounded open] subset o/R N . (i) The topologies induced by W^(D) (resp., Wljp(D}) on Cb(D) are all equivalent for p, 1 < p < oo. (ii) Cb(D) is closed in W^(D) (resp., W1'P(D)) forp, lp (D)-strong for all p, 1 < p < oo. (v) It is sufficient to prove the result for D bounded open. From part (iii) it is true for XdA, and from part (iv) and Theorem 4.1 (ii) and (iii) of Chapter 4 for the other two.
228
Chapter 5. Oriented Distance Function and Smoothness of Sets
Theorem 5.2. Let D be a bounded open domain in R N . (i) // {bAn} weakly converges in WI!P(D) for some p, 1 < p < oo, then it weakly converges in W1'P(D) for all p, 1 < p < oo. (ii) If {bAn} converges in C(D], then it weakly converges in Wljp(D) for all p, 1 < p < oo. Conversely, if {bAn} weakly converges in Wl'p(D] for some p, 1 < p < oo, it converges in C(D], (iii) Cb(D] is compact in Wljp(D}-weak for allp, 1 < p < oo. Proof, (i) Recall that for D bounded there exists a constant c > 0 such that for all bA £ Cb(D]
If {bAn} weakly converges in Wl'p(D) for some p > 1, then {bAn} weakly converges in LP(D), {V&A n } weakly converges in LP(D}N. By Lemma 2.1 (iii) in Chapter 3 both sequences weakly converge for all p > I and hence {bAn} weakly converges in W1'P(D] for all p>l. (ii) If {bAn} converges in C(D], then by Theorem 2.2 (ii) there exists bA € Cb(D) such that bAn -> bA in C(D) and hence in LP(D). So for all (p e T>(D)N
By density of V(D] in L2(D), VbAn -»• VbA in L2(jD)7v-weak and hence bAn -> bA in W1'2(D}-weak. From part (i) it converges in W1'p(D)-weak for all p, 1 < p < oo. Conversely, the weakly convergent sequence converges to some / in Wl'p(D). By compactness of Cb(D), there exist a subsequence, still indexed by n, and bA such that &An —* OA in C(D) and hence in W1'p(D}-weak. By uniqueness of the limit, bA = /• Therefore, all convergent subsequences in C(D] converge to the same limit. So the whole sequence converges in C(D). This concludes the proof. (iii) Consider an arbitrary sequence {6^n} in Cd(D). From Theorem 2.2 (ii) Cb(D] is compact and there exists a subsequence {bAn } and 6^ € Cb(D] such that \)An —> bA in C(D). From part (ii) the subsequence weakly converges in W1'P(D] and hence Cb(D) is compact in WAl>p(D)-weak.
6
Sets of Bounded and Locally Bounded Curvature
We introduce the family of sets with bounded or locally bounded curvature which are the analogues for 6^ of the sets of exterior and interior bounded curvature associated with dA and d^A m section 5 of Chapter 4. They include (71'1-domains, convex sets, and the sets of positive reach of Federer [2]. They lead to compactness theorems for Cb(D] in Wl>p(D).
6. Sets of Bounded and Locally Bounded Curvature
6.1
229
Definitions and Main Properties
Definition 6.1. (i) Given a bounded open nonempty subset D of R N , a subset A of D, dA ^ 0, is said to be of bounded curvature with respect to D if
This family of sets will be denoted as follows:
(ii) A subset A of R N , dA ^ 0, is said to be of locally bounded curvature if
where B(x,p) is the open ball of radius p > 0 in x. From Theorem 5.1 and Definition 5.1 in Chapter 3, a function belongs to BVi oc (R N ) if and only if for each x G RN it belongs to BV(B(x, p)} for some p > 0. As in Theorem 5.2 of Chapter 3 for Caccioppoli sets, it is sufficient to satisfy this condition for points of the boundary. Theorem 6.1. Let A, dA / 0, be a subset o/R N . Then A is of locally bounded curvature if and only ifVb^ belongs to BVi oc (R N )^ v . Corollary 2. All sets A, dA ^ 0, in RN of class C1'1 are of locally bounded curvature. The theorem will be proved as part (iii) of Theorem 6.3, and the corollary follows from Theorem 4.3 (i). Theorem 6.2. (i) Let D be a nonempty bounded open Lipschitzian subset of R N . For any subset A ofD, dA ^ 0, such that
dA has finite perimeter, that is,
(ii) For any subset A o/R N , dA ^ 0, such that
dA has locally finite perimeter, that is,
230
Chapter 5. Oriented Distance Function and Smoothness of Sets
Proof. Given V6^ in BV(D)N, there exists a sequence {uk} in C°°(D)N such that
as k goes to infinity, and since |V6>i(^)| < 1> this sequence can be chosen in such a way that This follows from the use of mollifiers (cf. Giusti [1, Thm. 1.17, p. 15]). For all V in T>(D)N
For each Uk
where *Duk is the transpose of the Jacobian matrix Du^ and
since for VF1>:1(Z))-functions, ||V/||£,i(£>)jv = HV/Ujv/^D)^- Therefore, as k goes to infinity,
Therefore, VXdA e Ml(D}N. Theorem 6.3. Let A be a subset o/R N such that dA ^ 0. (i) The function fdA(x) = \ (\x\2-b2A(x)) is convex mR N andVb2A e BVi oc (R N ) A For all x and y in RN
or equivalently
The set of projections Tig A (x) onto dA is a singleton PQA (x) if and only if the gradient oj JQA exists. In both cases
6. Sets of Bounded and Locally Bounded Curvature
231
For all x and y in RN
Vp(z) G iWz), Vp(y) G n aj4 (y),
(p(y) - p(x)) • (y - x) > 0.
(6.6)
The function b\(x] is the difference of two convex functions
(ii) VbA 6 BVi oc (R N \^) Jv . More precisely, for all x & RN \#A tfiere exists p > 0, 0 < 3/? < cfo^x), suc/i i/iai ^9^4^ is convex in B(x^2p) and hence VbAeBV(B(x,p))N. (ui) A subset A of R N , dA ^ 0, is of locally bounded curvature if and only if V^eBVioclR^. Proof. The proof follows from Lemma 5.1 and Theorem 5.3 in Chapter 4 by applying it to A\- Use the fact that bA — dA and bA = — d^A outside of dA. 6.2
Examples and Limits of the Tubular Norms as h Goes to Zero
It is informative to compute the Laplacian of bA for a few examples. The first three are illustrated in Figure 5.1.
Figure 5.1. VbA for Examples 6.1, 6.2, and 6.3. Example 6.1 (half-plane in R2; cf. Example 5.1 in Chapter 4). Consider the domain
It is readily seen that
and
232
Chapter 5. Oriented Distance Function and Smoothness of Sets
Example 6.2 (ball of radius R > 0 in R2; cf. Example 5.2 in Chapter 4). Consider the domain
Clearly,
Also
Again AbdA contains twice the boundary measure on dA. Example 6.3 (unit square in R2; cf. Example 5.3 in Chapter 4). Consider the domain
Since A is symmetrical with respect to both axes, it is sufficient to specify 6^ in the first quadrant. We use the notation Qi, Q^, Qs, and Q± for the four quadrants in the counterclockwise order and c\, C2, 03, and 04 for the four corners of the square in the same order. We also divide the plane into three regions:
Hence for 1 < i < 4
and for the whole plane
6. Sets of Bounded and Locally Bounded Curvature
233
where D\ n D^ is made up of the two diagonals of the square where V6^ has a singularity, that is, the skeleton. Moreover, for 1 < i < 4
and
Notice that the structures of the Laplacian are similar to the ones observed in the previous examples except for the presence of a singular term along the two diagonals of the square. All C2-domains with a compact boundary belong to all the categories of Definition 6.1 and Definition 5.1 of Chapter 4. The /i-dependent norms \\D2bA\\M1(uh(dA))i \\D2d£A\\Mi(UhfiA^, and \\D2dA\\Mi(uh(A)) are au decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary dA.
Example 6.4.
If A C RN is of class C2 with compact boundary, then
Example 6.5.
Let A = {xi}f_1 be / distinct points in R N . Then dA = A and
Example 6.6.
Let A be a closed line in RN of length L > 0. Then dA — A, and
Example 6.7. Let N = 2. For the finite square and the ball of finite radius,
where HI is the one-dimensional Hausdorff measure (cf. Examples 5.2 and 5.3 in Chapter 4).
234
Chapter 5. Oriented Distance Function and Smoothness of Sets
Also, looking at A6^ in Uh(A) as h goes to zero provides information about Sk(A).
Example 6.8.
Let A be the unit square in R2. Then
where HI is the one-dimensional Hausdorff measure and Sk(A) = Skjnt(A) is the skeleton of A made up of the two interior diagonals (cf. Example 6.3). It would seem that in general
where [V&AJ is the jump in V&A and n is the unit normal to Sk(^4) (if it exists!). D
6.3
(m,p)-Sobolev Domains (or Wm'p-Domains)
The TV-dimensional Lebesgue measure of the boundary of a Lipschitzian subset of RN is zero. However, this is generally not true for sets of locally bounded curvature, as can be seen from the following example.
Example 6.9. Let B be the open unit ball centered in 0 of R and define A = [x 6 B : x with rational coordinates}. Then dA = £, bA = dB, and for all h > 0
The next natural question that comes to mind is whether the boundary or the skeleton have locally finite (N — l)-Hausdorff measure. Then come questions about the mean curvature of the boundary. For the function 6^, A6^ = trD 2 6^ is proportional to the mean curvature of the boundary, which is only a measure in RN for sets of bounded local curvature. This calls for the introduction of some classification of sets that would complement the classical Holderian terminology introduced in Chapter 2, would fill the gap in between, arid would possibly say something about sets whose boundary is not even continuous. This seems to have been first introduced in Delfour and Zolesio [32].
Definition 6.2 (Sobolev domains).
Given m > 1 and p > 1, a subset A of RN is said to be an (m,p)-Sobolev domain or simply a Wm'p-domain if dA ^ 0 and there exists h > 0 such that
6. Sets of Bounded and Locally Bounded Curvature
235
This classification gives an analytical description of the smoothness of boundaries in terms of the derivatives of bA. The definition is obviously vacuous for m = 1 since bA G ^'c°°(RN) for any set A such that dA / 0. For 1 < m < 2 we shall see that it encompasses sets that are less smooth than sets of locally bounded curvature. For m — 2 the VT2'p-domains are of class CI:I~N/P for p > N. For m > 2 they are intertwined with sets of class Ck^. Theorem 6.4. Given any subset A o/R N , dA ^ 0,
Proof. Since \VbA(x)\ < I almost everywhere, VbA G BVi oc (R N ) jV n L£ C (R N ) N and the theorem follows directly from Theorem 5.8 in Chapter 3. It is quite interesting that a domain of locally bounded curvature is a W2~£^domain for any arbitrary small e > 0. So it is almost a VT^-domain, and domains of class W2~£'1 seem to be a larger class than domains of locally bounded curvature. In addition, their boundary does not generally have zero measure. Now consider the boundary case m = 2. Theorem 6.5. Given an integer N > I , let A be a subset o/R N such that dA ^ 0. (i) // there exist p > N and h > 0 such that
then
that is, A is a Holderian set of class Cl'l~N/p. Moreover, the function b2A belongs to Clo'c (Uh(dA)), dA has positive reach, and the N-dimensional Lebesgue measure of dA is zero. (ii) In dimension N = 2 the condition bA G Wj 'p(Uh(dA}) is equivalent to AbA £ Llc(Uh(dA}}. Proof, (i) Consider the function VbA\2. Since \VbA\ < 1, then
In particular, for all
236
Chapter 5. Oriented Distance Function and Smoothness of Sets
For p > TV, from Adams [1, Thm. 5.4, Part III, and Remark 5.5 (3), p. 98],
From Theorem 4.1, b\ e C^(Uh(dA)) and dA has positive reach. From Theorem 4.2 A is of class Cl>l~N'p and m(dA) = 0. (ii) From part (i), |V6^(o;)|2 = 1 in Uh(dA), and D2bA(x) VbA(x) = 0 almost everywhere. Hence in dimension N = 2,
As can be seen, a certain amount of work would be necessary to characterize all the Sobolev domains and answer the many associated open questions.
7
Characterization of Convex and Semiconvex Sets
We have seen in Chapter 4 that the convexity of dA is equivalent to the convexity of A. We shall see that this characterization remains true with b^ in place of dA — dA. However, the next example shows that the convexity of bA is not sufficient to get the convexity of A.
Example 7.1.
Let B be the open unit ball in RN and A the set B minus all the points in B with rational coordinates. By definition,
By Theorem 6.2 (i) of Chapter 4 the function d,B and, a fortiori bA, are convex, but A is not convex and dA ^ dA.
Theorem 7.1.
7. Characterization of Convex and Semiconvex Sets
237
(iii) Let A be a subset o/R N such that dA ^ 0. Then
(iv) For all convex sets A such that dA ^ 0, VbA belongs to BVi oc (R N ) 7V and the Hessian matrix D2bA of second-order derivatives is a matrix of signed Radon measures which are nonnegative on the diagonal. Moreover, bA has a second-order derivative almost everywhere, and for almost all x and y in R N ,
Proof, (i) Denote by x\ the convex combination Ax + (1 — X)y of two points x and y in A for some A G [0,1]. By convexity of bA,
and A is convex. (ii) It is sufficient to prove the result for A closed. In the first direction the result follows from part (i). In the other direction we consider three cases. The first one deserves a lemma. Lemma 7.1. Let A be a subset o/R N such that dA ^ 0 and A be convex. Then
Proof. Again we can assume that A is closed. Associate with x and y in ,4, the radii rx = d,£A(x}, ry = d^A(y], r\ = \rx + (1 — A)r y , and the closed balls Bx of center x and radius r x , By of center y and radius ry, and B\ of center x\ and radius r\. By the definition of C?QA, Bx C A and By C A since A is closed. Associate with each z € B\ the points zx = x and zy — y if r\ = 0, and if r\ > 0 the points
Obviously z = x\ if r\ = 0. Therefore,
since A is closed and convex. In particular
and this proves the lemma.
238
Chapter 5. Oriented Distance Function and Smoothness of Sets
For the second case consider x and y in CA By definition
By Theorem 6.2 (i) of Chapter 4, d,A is convex when A is convex and
The third and last case is the mixed one for x £ CA. and y (~ A. Define
Since A is convex, denote by H the tangent hyperplane to A through p\, by H+ the closed half-space containing A, and by H~ the other closed subspace associated with H. By the definition of H,
and by convexity of A, A C H+ and H~ C CA The projection onto H is a linear operator and
By convexity any y belongs to A C H+ and since H
C CA, we readily have
First consider the case x\ G CA. Then d,A(x\) > 0, x\ 6 H~, and necessarily xeH~. From (7.5)
But since x € H~ and A C H+
and
7. Characterization of Convex and Semiconvex Sets
239
Next consider the case x\ G A for which x\ G H+ and If x G H~, then since A C H+,
and from (7.5)
If, on the other hand, x G ff + , then x, y G #+ and XA G A C # + . So from (7.4)
This covers all cases and concludes the proof. (iii) Consider two cases. For any convex set with a nonempty interior, int A = int A and hence CCA = CCA. Therefore, CA = CA and d^A = d^A. For a convex set with an empty interior, A = p(D], 1 < p < oo. Moreover,
Proof. It is sufficient to prove the result for D bounded. Furthermore, by compactness of Cb(D] in C(D), it is sufficient to prove that Cb(D) and Cb(D;E) are closed in C(D}. (i) Let {bAn} be a Cauchy sequence in Cb(D). It converges to some b^ £ Cb(D). By Theorem 7.1 (iii), the &A n ' s are convex and so is the limit 6^, and by Theorem 7.1 (i), A is convex, and by Theorem 7.1 (ii), b^ is convex. Consider the following two cases: int A = 0 and int A ^ 0. (1) In the first case
Therefore
and b&n —> ^A — bA, where A is convex: bA — b^ e Cb(D}. (2) For the second case, intA^0 and A convex imply that int A = A by Theorems 5.4 and 5.3 in Chapter 2. Consider the following two subcases: int A = A and int A ^ A. (2a) If int A = int A = A, then, using the fact that Cint A — C^4, bint A — dint A ~ ^Cint A ~ °^A ~ d$A = 6^,
and dint A = dA ^ 0. Thus 6^n —> b^ = &mtA> where int A is convex by Theorem 6.2 (ii). (2b) We show that int^l = A leads to a contradiction and cannot occur. If int A ^ A, then either int A = 0 or int A ^ 0. If int A = 0, then dA = A, and since int A ^ 0, there exist x and r > 0 such that B(x,r) C dA. If mtA^0
7. Characterization of Convex and Semiconvex Sets
241
is open (convex) and not empty. Moreover, since B(a, d) n int A = 0,
and there exists a ball B(x,r), r > 0, such that B(x,r) C dA. Thus in both cases there exist x and r > 0 such that J5(x,r) C dA. We prove that this leads to a contradiction. Choose N such that for all n > N
We shall prove that for all n > N there exists xn G B(x,r] such that dAn(xn) > 3r/8. Hence xn G B(x,r) C dA C A and from (7.8)
which yields a contradiction and proves that necessarily int A = A, and we are back to case (2a). Let pn G dAn be a projection of x onto dAn and choose
since An is convex and bounded. Choose
Let
H+ d= {y € RN : (y -Pn) • vn > 0} , H~ d= {y G RN : (y -
Pn)
• vn < 0} .
We claim that An C Hn . This is obvious if x G CA n . For x G intA n we can show by contradiction that if there exists z G H+ such that z E An, then (z— pn) -vn > 0, and since B(x,d^An^(x}) C An
From this it is easy to show that pn G int An. But this would contradict the fact that by definition pn G dAn. As a result
By definition of H+, xn G H+ and
242
Chapter 5. Oriented Distance Function and Smoothness of Sets
However, An C Hn and necessarily
and this concludes the proof of part (i). (ii) Prom part (i), given a Cauchy sequence {bnn} in Cb(D;E), there exists a convex subset A of D such that 6^ —» bA. Moreover, since d^n —»• d$A, E C fin => C£ D C0n =*> C£ D Cl - Cint A => 0 ^ E C int A Since ^4 is convex and int A ^ 0, int A — A and 6^4 = 6int A- Thus O = int A can be chosen as the limit set. (iii) Let {bAn} C Cb(D] be _a Cauchy sequence in W1'P(D). From Theorem 5.2 (ii) it is also Cauchy in C(D). Hence from part (i) there exist a convex set A, dA i- 0, such that bAn -> bA in C(D] and, a fortiori, in W1'P(D). This shows that Cb(D] is closed in Wl>p(D}. To show that it is compact consider a sequence {bAn} C Cb(D}. From part (i) there exists a subsequence, still indexed by n, and a convex subset A, dA ^ 0, of D such that bAn —> 6^ in C(D). Since D is bounded it also converges in LP(D}. Since |V6^n| is pointwise bounded by one, there exists another subsequence of {bAn}, still indexed by n, such that V6^n converges to VbA in Wl-p(D)N-weak. But An and A are convex. Thus m(dA) = 0 = m(5^n) and |V&A n | = 1 = |V6yi| almost everywhere in R N . As a result, for p = 2
Hence we get the compactness in W^1;2(.D)-strong and by Theorem 5.1 (i) in Wljp(D)strong p, 1 < p < oo. (iv) The result follows from the continuity of the map (5.5) in Theorem 5.1 (iv). We have seen in Theorem 7.1 (iv) that in view of the convexity of the distance (resp., oriented distance) functions of a closed convex set A, closed convex sets are of locally bounded exterior curvature and of locally finite perimeter (resp., locally of bounded curvature). When A is convex and of class C2, then for any X e dA, there exists a strictly convex neighborhood N(X) of X such that
or, since D2bA(x) VbA(x) = 0 in N(X), then for all x € N(X), This is related to the notion of strong elliptic midsurface in the theory of shells: there exists c > 0 All this motivates the introduction of the following notions.
8. Federer's Sets of Positive Reach
243
Definition 7.1. Let A be a closed subset of RN such that dA ^ 0. (i) The set A is locally convex (resp., locally strictly convex) if for each X G dA there exists a strictly convex neighborhood N(X) of X such that
(ii) The set A is semiconvex if
(iii) The set A is locally semiconvex if for each X S dA there exists a strictly convex neighborhood N(X] of X and
Remark 7.1. When a closed set A has a compact C2 boundary, D2bA is bounded in a bounded neighborhood of 0 such that VbA e BV(C/'/ l (9A)) N . Remark 7.2. Given a fixed constant /3 > 0, consider all the subsets of D that are semiconvex with constant 0 < a < (3. Then this set is closed for the uniform and the V^llp-topologies, 1 < p < oo.
8
Federer's Sets of Positive Reach
The distance function dAr of the dilated set Ar = {x € RN : dA(x) < r}, r > 0, of A provides a uniform approximation of cU, as can be seen from the following theorem. Theorem 8.1. Assume that A is a nonempty subset o/R N . For r > 0 and, as r goes to zero,
Proof. For any q e Ar, d,A(q) < r, and for p e H^(g),
When, in addition, A has positive reach, reach(A) > r, and Ar ^ RN for some r > 0, the dilated sets Ar are of class C1'1 and 6^ can be approximated by bAr in the W^-P-topology.
244
Chapter 5. Oriented Distance Function and Smoothness of Sets
Theorem 8.2. Let A be a nonempty subset o/R N . Assume that there exists h > 0 such that (i) For all r, 0 < r < h, dAr ^ 0 and Ar is a set of class C1'1 such that
and for each x £ dAr there exists a neighborhood V(x) of x in Uh(A)\A such that
(ii) Moreover, dA ^ 0, and for all r, 0 < r < h,
In particular,
(iii) As r goes to zero, 6^r —> b^ uniformly in R N ; and for p > I ,
(iv) Furthermore, A is of local bounded curvature and dA has zero volume
The proof of the theorem requires the following two lemmas. Lemma 8.1. Assume that A is a nonempty subset o/R N . For r > 0 such that Ar^RN
Proof of Lemma 8.1. (i) First inequality. For all y such that cU(y) > f and all P£ A,
8. Federer's Sets of Positive Reach
245
Second inequality. Given x G A, by definition of d^A(x]^ Bx = B(x,d^(x}} is a subset of A. For r > 0 there exists pr £ [>Ar such that
Define
By construction,
and d^Ar(x] > d^(x] + r. Lemma 8.2. Let A be a nonempty subset o/R N and assume that there exists h > 0 such that d\ e C^(Uh(A)). For all x e C// l (A)\A and 0 < r < h,
Proof of Lemma 8.2. In the region Uh(A)\A the gradient Vd/i is locally Lipschitz. For any point x, 0 < dA(x) < h, consider the flow
There exists a unique local solution through x. Moreover,
Therefore, the solution exists and is unique for s, 0 < s < h — p. By Lemma 8.2 and part (i), associate with each yn
for which pA(Qn) — PAfajn)- By Theorem 7.1 in Chapter 4, p^ belongs to (^{^(^(A)). In particular, PA G C 0ll (5(p, (r + h}/2)} and there exists c > 0 such that
Since {qn} is bounded there exists q and a subsequence, still indexed by n, such that
248
Chapter 5. Oriented Distance Function and Smoothness of Sets
Therefore,
Henceforth, for each x £ A we can associate with p £ Ila^(x) a q 6 dAr such that PA(Q) =P and
From the second inequality in Lemma 8.1,
and since dA(x) = 0 = dAr(x),
(iii) The uniform convergence of bAr to bA in RN is a direct consequence of part (ii). For all bounded open subsets D of Uh(A) as r —> 0,
since from part (ii), VbAr = V6^ in Uh(A). The convergence of dAr = &J[ , d^Ar = bAr, and rfaAr = I&/U to 6± = dA, 6~ = d c ^, and \bA\ = ddA in W^t/^A)) is now a consequence of Theorem 5.1 (iv). (iv) From part (i) for all 0 < r < h, Ar is a set of class C1'1, and for each x € dAr there exists a neighborhood V(x) of a; in Uh(A)\A such that 6^r G C 1 ' 1 (V r (x)). By Theorem_6.1, VbAr 6 BVi oc (R N ) JV . Therefore, by Definition 5.1 in Chapter 3, for all x € dA,
since from part (ii) V6^r = VbA in Uh(A). By Theorem 6.1 (ii), A is of locally bounded curvature and, by Theorem 6.1, in BVi oc (R- N ) JV - Finally, by continuity of the map
9
Compactness Theorems for Sets of Bounded Curvature
For the family of sets with bounded curvature, the key result is the compactness of the embedding
9. Compactness Theorems for Sets of Bounded Curvature
249
for bounded open Lipschitzian subsets of RN and p, 1 < p < oo. It is the analogue of the compactness Theorem 5.3 of Chapter 3 for Caccioppoli sets
which is a consequence of the compactness of the embedding
for bounded open Lipschitzian subsets of RN (cf. Morrey [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and Evans and Gariepy [1, Thm. 4, p. 176]). As for characteristic functions in Chapter 3, we give a first version involving global conditions on a fixed bounded open Lipschitzian hold-all D. In the second version the sets are contained in a bounded open hold-all D with local conditions in the tubular neighborhood of their boundary.
9.1
Global Conditions on D
Theorem 9.1. Let D be a nonempty bounded open Lipschitzian subset o/R N . The embedding (9.1) is compact. Thus for any sequence {An}, dAn ^ 0, of subsets of D such that
there exist a subsequence {Ank} and a set A, dA ^ 0, such that VbA e EV(D}N and
Proof. Given c > 0 consider the set
By compactness of the embedding (9.3), given any sequence {bAn} there exist a subsequence, still denoted {bAn}, and / e EV(D)N_such that VbAn -> / in L1(D}N. But by Theorem 2.2 (ii), Cb(D) is compact in C(D) for bounded D and there exist another subsequence {bAn } and bA £ Cb(D} such that bAn —> bA in C(D) and, a fortiori, in Ll(D}. Therefore, bAn converges in Wljl(D] and also in Ll(D). By uniqueness of the limit, / = VbA and bAn converges in W1>l(D} to bA. For $ G Vl(D)NxN as k goes to infinity
250
Chapter 5. Oriented Distance Function and Smoothness of Sets
||-^)2^A||M1(D) < ci and V6^ G BV(D)N. This proves the sequential compactness of the embedding (9.1) for p = 1 and properties (9.5). The conclusions remain true for P > 1 by the equivalence of the W^lip-topologies on Cb(D] in Theorem 5.1 (i). D
9.2
Local Conditions in Tubular Neighborhoods
The global condition (9.4) is now weakened to a local one in a neighborhood of each set of the sequence. Simultaneously, the Lipschitzian condition on D is removed since only the uniform boundedness of the sets of the sequence is required. Theorem 9.2. Let D be a nonempty bounded open subset o/R N and {An}, 0 ^ dAn, be a sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that
Then there exist a subsequence {Ank} and a subset A, 0 ^ dA, of D such that V&A e BVioc(R N ) jV , and for allp, l°(Uh(dA)),
and XdA belongs to BVi oc (R N )Proof, (i) By assumption, An C D implies that Uh(An) C Uh(D). Since Uh(D) is bounded, there exist a subsequence, still indexed by n, and a subset A of D, dA ^ 0, such that
and another subsequence, still indexed by n, such that
For all e > 0, 0 < 3e < h, there exists N > 0 such that for all n > N and all x € Uh(D),
Therefore,
9. Compactness Theorems for Sets of Bounded Curvature
251
From (9.6) and (9.10) In order to use the compactness of the embedding (9.3) as in the proof of Theorem 9.1, we would need Uh-e(dA) to be Lipschitzian. To get around this we construct a bounded Lipschitzian set between Uh-2e(dA) and Uh-e(dA). Indeed by definition, and by compactness there exists a finite sequence of points {xi}"=1} in dA such that
Since UB is Lipschitzian as the union of a finite number of balls, it now follows by compactness of the embedding (9.3) for UB that there exists a subsequence, still denoted {bAn}, and / G BV(UB)N such that VbAn -> / in Ll(UB)N. Since Uh(D) is bounded, Cb(Uh(D}} is compact in C(Uh(D}} and there exists another subsequence, still denoted {bAn}, and A C D, dA ^ 0, such that bAn —> bA in C(Uh(D}) and, a fortiori, in Ll(Uh(D}}. Therefore, bAn converges in W1'1(C/s) and also in Ll(Us)- By uniqueness of the limit, / = VbA on UB and bAn converges to bA in Wl'l(UB}. By Definition 6.1 and Theorem 6.3 (iii), V6A and V6An all belong to BVioc(RN)]V since they are of bounded variation in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 6.2 (ii), XdA G BVioc(RN). The above conclusions also hold for the subset Uh-2e(dA) of UB(ii) Convergence in Wl'p(Uh(D)}. Consider the integral
Prom part (i) the first integral on the right-hand side converges to zero as n goes to infinity. The second integral is on a subset of ^>Uh-ie(dA). From (9.11) for all n>N,
The second integral reduces to
which converges to zero by weak convergence of VbAn to VbA in the space L2(Uh(D)) in part (i) and the fact that \VbA — I almost everywhere in Uh(D}\Uh-is(dA}. Therefore, since bAn —> bA in C(Uh(D}}
252
Chapter 5. Oriented Distance Function and Smoothness of Sets
and by Theorem 5.1 (i) the convergence is true in W1>p(Uh(D)) for all p > I. (Hi) Properties (9.8). Consider the initial subsequence {bAn} which converges to bA in H1(Uh(D))-weak constructed at the beginning of part (i). This sequence is independent of E and the subsequent constructions of other subsequences. By convergence of bAn to bA in Hl(Uh(D))-weak for each $ e T>1(Uh(dA))NxN,
Each such ) > 0, 0 < 3e < h, such that From part (ii) there exists N(s) > 0 such that
For n > N(e) consider the integral
By convergence of VbAn to VbA in the space L2(Uh(D))-weak, then for all 3> 6
Vl(Uh(dA)}NxN
Finally the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.
10
Compactness and Uniform Cone Property
In Theorem 5.9 of section 5.4 in Chapter 3, we have seen a compactness theorem for the family of subsets of a bounded hold-all D satisfying the uniform cone property. In this section we give a direct proof of the compactness for the C(D}- and Wl'p(D)topologies associated with the oriented distance function b^. As a consequence we get the compactness for the C(D}- and W/1'p(Z))-topologies associated with d^ and d,£ft. Furthermore, we recover the compactness of Theorem 5.9 of Chapter 3 for Xfi in LP(D). Finally we also get the compactness for xcn m LP(D}. Recall the following notation and definition
10. Compactness and Uniform Cone Property
253
We have seen in Theorem 6.1 of Chapter 2 that for a Lipschitzian set fi, 90 ^ 0, m( 0, u; > 0, and A > 0, t/ie family
is compact in C(D] and Wl'p(D}. As a consequence the families
are compact in C(D] and W 1)P (D), and the families
are compact in LP(D). The proof of this theorem is similar to the proof of Theorem 5.9 in Chapter 3 and uses a key lemma. Proof of Theorem 10.1. (i) Compactness in C(D}. Consider an arbitrary sequence {fin} in L(D,r,u,\). For D compact Cb(D) is compact in C(D) and there exists 0 C D and a subsequence {Onfc} such that
It remains to prove that 0 e L(D, r, a;, A). This requires the equivalent of Lemma 5.1 in Chapter 3. Lemma 10.1. Given a sequence {ban} C Cb(D] such that b^n —> 6^ in C(D] for some b$i 6 Cb(D), we have the following properties:
and for all x £ CO,
254
Chapter 5. Oriented Distance Function and Smoothness of Sets
Moreover,
and Proof. We proceed by contradiction. Assume that
So there exists a subsequence {J7nfc}, n& —•> oo, such that
which contradicts the fact that 6nn —> &n- Of course, the same assertion is true for the complements and for all x G CO,
when x € 50, x € 0 fl CO, and we combine the two results. For the last result use the fact that the open ball cannot be partitioned into two nonempty disjoint open subsets. Coming back to the proof of the theorem, we wish to show that 0 has the uniform cone property
Since On is Lipschitzian, so is COn, and from the second part of the lemma for each x G 5O, Vfc > 1, 3nk > k such that
Denote by Xk an element of that intersection:
By construction Xk —> x. Next consider y e B(x, r) D 0. From the first part of the lemma, there exists a subsequence of {Onfc}, still denoted {Onfc}, such that
For each k > 1 denote by yk a point of that intersection. By construction
10. Compactness and Uniform Cone Property
255
There exists K > 0 large enough such that
To see this, note that y G B(x, r) and that
Now
Since r/p > I the result is true for
So we have constructed a subsequence {rinfc} such that for k > K
For each fc, 3dfc G R N , \dk = 1, such that
Pick another subsequence of {f2 nfc }, still denoted {f£ nfc }, such that
Now consider z G C y (A,u;,d). Since z is an interior point
and there exists K' > K such that
This proves that £7 C D satisfies the uniform cone property and f) G L(D, A,o;,r). (ii) Compactness in Wl'p(D). From Theorem 5.1 (i) it is sufficient to prove the result for p = 2. Consider the subsequence {Qnk} C L(D, A,o;,r) and let f2 G L(D, A,o;,r) be the set previously constructed such as b$in —» 6^ in C(D). Hence 6^n -^ b& in L2(D}. Since D is compact for all fi C D
256
Chapter 5. Oriented Distance Function and Smoothness of Sets
and there exists a subsequence, still denoted {6^n }, which converges weakly to 6^. Since all the sets are Lipschitzian, m(, d) is replaced by the segment (0, Xd). This is readily seen by considering the following example. Example 11.1. Given an integer n > 1, consider the following sequence of open domains in R :
They satisfy the uniform segment property of Definition 7.1 (ii) of Chapter 2 by choosing A = r = 1/4. The sequence {fin} converges to the closed set
in the uniform topologies associated with d^n and ban or in the I/p-topologies associated with xnn and Xcnn • However, the segment property is not satisfied along the line L and the corresponding family of subsets of the hold-all D — B(Q, 4) satisfying the uniform segment property with r = A = 1/4 is not closed and, a fortiori, not compact.
11. Compactness and Uniform Cusp Property
257
This example shows that a uniform segment property is too meager to make the corresponding family compact. Looking back at the proof of Theorem 10.1, everything goes through with the uniform segment property except in the last seven lines of the proof of part (i), where the fact that the cone is an open set is critically used. This suggests that the cone could be replaced by an open cusp or horn that would yield larger families than those of Lipschitzian domains considered in the previous section. For instance, the cone C(A,o;,d) can be replaced by the cuspidal region
where B#(0,p) is the open ball of radius p > 0 and center 0 in the hyperplane H through 0 orthogonal to the direction d and h : [0, p] —->• R is a continuous function such that
This will be referred to as a uniform cusp property. Hence, we can now choose functions h : [0, p] —>• R of the form
The special case of the cone corresponds to
The following theorem is now a corollary to Theorem 10.1. Theorem 11.1. Let the assumptions of Theorem 10.1 be verified with the cone C(\,ui,d) replaced by the cuspidal region H(X,h,d) defined in (11.1) under the conditions (11.2) on the continuous function h. Then the compactness properties of Theorem 10.1 remain true. In some applications, it might be interesting to relax the uniform cusp property by permitting the axis of the cuspidal region to bend: this makes the region look like a horn and the corresponding property becomes a horn condition or property. Horn-shaped domains have been studied in several contexts in the literature. In particular, conditions on domains have been introduced in the context of extension operators and embedding theorems: the domains of F. John;2 the (e, 5)-domains of P. W. John;3 and domains satisfying a flexible horn condition (which is a broader notion than the previous two) by O. V. Besov.4 2
F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391-413. P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88. 4 O. V. Besov, Integral representations of functions in a domain with the flexible horn condition, and embedding theorems (Russian), Dokl. Akad. Nauk SSSR 273 (1983), 1294-1297; English translation: Soviet Math. Dokl. 28 (1983), 769-772. O. V. Besov, Embeddings of an anisotropic Sobolev space for a domain with a flexible horn condition (Russian), in "Studies in the Theory of Differentiable Functions of Several Variables and Its Applications, XII," Trudy Mat. Inst. Steklov 181 (1988), 3-14; 269; English translation: Proc. Steklov Inst. Math. (1989), 1-13. 3
This page intentionally left blank
Chapter 6
Optimization of Shape Functions
1
Introduction and Generic Examples
In Chapter 3 we have seen several examples of optimization problems involving the Lp-topology on measurable sets via the characteristic function. In this chapter we consider optimization problems where the underlying topology is specified by the distance functions of Chapters 4 and 5. We give a brief account of two generic optimization problems: the minimization of a quadratic objective function which depends on the solution of the homogeneous Dirichlet boundary value problem associated with an elliptic operator, and the optimization of the first eigenvalue of the same operator. In both problems the strong continuity of solutions of the elliptic equation or the eigenvector equation with respect to the underlying domain is the key element in the proof of the existence of optimal domains. To get that continuity, some extra conditions have to be imposed on the family of open domains. Those questions have received a lot of attention for the Laplace equation with homogeneous Dirichlet boundary conditions. To a sequence of domains in a fixed hold-all D, associate a sequence of extensions by zero of the solutions in the fixed space HQ(D}. The classical Poincare inequality is uniform for that sequence as the first eigenvalue of the Laplace equation in each domain is dominated by the one associated with the larger hold-all D. By a classical compactness argument, the sequence of extensions converges to some limiting element y in HQ(D). To complete the proof of the continuity, two more fundamental questions remain. Is y the solution of the Laplace equation for the limit domain? Does y satisfy the Dirichlet boundary condition on the boundary of the limit domain? The first question can be resolved by assuming that the Sobolev spaces associated with the moving domains converges in the Kuratowski sense, i.e., with the property that any element in the Sobolev space associated with the limit domain can be approached by a sequence of elements in the moving Sobolev spaces. That property is obtained in examples where the compactivorous property1 follows from the choice of 1
A sequence of open sets converging to a limit open set in some topology has the compactivorous property if any compact subset of the limit set is contained in all sets of the sequence after a certain rank: the sequence eats up the compact set after a certain rank.
259
260
Chapter 6. Optimization of Shape Functions
the definition of convergence of the domains. If the domains are open subsets of D, then the complementary Hausdorff topology has that property. For the second question it is necessary to impose constraints at least on the limit domain. The stability condition introduced by Rauch and Taylor [1] and used by Dancer [1] and Daners [1] precisely assumes that the limit domain is sufficiently smooth, just enough to have the solution in HQ (Q). Of course assuming such a regularity on the limit domain makes things easier. The more fundamental issue is to identify the families of domains for which this stability property is preserved in the limit. The uniform cone property has been used in the context of shape optimization by Chenais [1, 2] in 1973. Capacity conditions have been introduced in 1994 by Bucur and Zolesio [3, 5, 6, 8, 9] in order to construct compact subfamilies of domains with respect to the Hausdorff complementary topology (see section 2.3 of Chapter 4). Furthermore, Bucur [1] proved that the condition given in 1993 by Sverak [1, 2] in dimension 2 involving a bound on the number of connected components of the complement of the domain can be recovered from the more general capacity conditions that are sufficient in the case of the Laplacian with homogeneous Dirichlet boundary conditions. In a more recent paper, Bucur [5] proved that they are almost necessary. Intuitively those capacity conditions are such that, locally, the complement of the domains in the family under consideration has enough capacity to preserve and retain the homogeneous Dirichlet boundary condition in the limit. As a first example consider the minimization of the objective function
over the family of open subsets 17 of a bounded open hold-all D of R N , where g in L2(D), un € HQ($I) is the solution of the variational problem
A e L°°(D; £(R N , R N )) is a matrix function on D such that
for some coercivity and continuity constants 0 < a < /?, and /|n denotes the restriction of / in H~l(D] to H~1(tt). The second example is the optimization of the first eigenvalue of the associated differential operator
We shall not provide a parallel treatment of the above problems for the homogeneous Neumann boundary conditions. However, the techniques are similar and most results remain true. In section 5 we introduce the fundamental //^-density
2. Embedding of #) in H$(D). Thence $ £ H$(0; D}. (ii) Since f2 is bounded, there exists a sufficiently large open ball B such that Q C B. By the embedding of HQ(^} into HQ(B], if a sequence (f>n converges to (f> in #Q (fi)-weak, then e^((pn] converges to e.$i((p} in HQ (B)-weak. Since the ball is sufficiently smooth, by Rellich's theorem, the sequence converges in H0~~l(B)-strong, and in view of the linear isometric isomorphism, (f>n converges to (f> in ^Q~ 1 (^)-strong. D
262
Chapter 6. Optimization of Shape Functions Going back to the restriction of an element / € H~l(D] to 17 c .D, define
which makes sense since
and by density and continuity /(^ readily extends to a continuous linear form on the subspace jF/g(17;.D) of HQ(D) and a fortiori on #0(17). Furthermore, the homogeneous Dirichlet boundary value problem (1.2) in 17 is completely equivalent to the variational problem: to find u £ HQ(^; D) such that
and u = efi(ufi). Define the following continuous symmetrical linear operator on D and its restriction to 17:
With the above notation, equation (1.2) reduces to
The following technical lemma will be useful. Theorem 2.2. Given a bounded open domain 17 in R N , the minimization problem (1.4) has nonzero solutions in H(j(£l) which are solution of the eigenvector equation
Given a bounded open nonempty domain D, there exists A^ > 0 (which only depends on the diameter of D and A) such that for all open subsets 17 of D,
Proof, (i) For any bounded open 17, the infimum is bounded below by 0 and hence finite. Let {(pn} be a minimizing sequence such that ||, A) such that
The conclusions remain true when L(D,r, u;, A) is replaced by CE(D) defined in (6.1). Proof. This follows from Theorem 6.1, where it is shown that the convergence is strong not only in the Hausdorff topologies, but also for the associated characteristic functions in Ll(D], thus making the volume function continuous with respect to open domains in L(D,r,u;, A). For the case of CE(D), recall from Theorem 7.2 (iii) of Chapter 5 that Cb(D; E] is compact in Wl'1(D), and hence the volume functional is continuous.
7
Elements of Capacity Theory
We have seen in the previous section that the key element in the proof of the continuity of the solution of (1.2) with respect to its underlying domain is that the limiting element satisfies the homogeneous Dirichlet boundary condition for the limiting domain. This section introduces capacity constraints which generalize the property to a broad class of open domains.
7.1
Definition and Basic Properties
The following definition of capacity and a number of basic technical results can be found in Hedberg [1] and are summarized below.
7. Elements of Capacity Theory
273
Definition 7.1. Let D be a fixed bounded open subset of R N . (i) The capacity2 (with respect to D] is defined as follows: — for a compact subset K C D
— for an open subset G C D
— for an arbitrary subset E C D
(ii) A function / on D is said to be quasi-continuous if
(iii) A set E in RN is said to be quasi-open if
It can easily be shown that for a quasi-open set there exists a decreasing sequence {£ln} of open sets, such that On D E and c&pD(£ln\E) goes to zero as n goes to infinity. We say that a property holds quasi-every where (q.e.) in D if it holds in the complement D\E of a set E of zero capacity. A set of zero capacity has zero measure, but the converse is not true. The capacity is a countably subadditive set function, but it is not additive even for disjoint sets. Hence the union of a countable number of sets of zero capacity has zero capacity.
7.2
Quasi-Continuous Representative and H^Functions
Lemma 7.1 (Hedberg [1]). Any f in Hl(D] has a quasi-continuous representative: there exists a quasi-continuous function f i defined on D such that /i = / almost everywhere in D (hence f i is a representative of f in H1(D}). Any two quasi-continuous representatives of the same element of Hl(D) are equal quasieverywhere in D. The following key lemma completes the picture. 2
This definition is also referred to as the exterior capacity.
274
ChapterG. Optimization of Shape Functions
Lemma 7.2 (Hedberg [1]). Lettl andD be two bounded open subsets o/R N such that Q C D and consider an element u of HQ(D). Then u\n 6 #o(^) tf ana onty if Ui = 0 quasi-every where on D\£l, where u\ is a quasi-continuous representative ofu. A function in Hl(D] is said to be zero quasi-everywhere in a subset E of D if there exists a quasi-continuous representative of (p which is zero quasi-everywhere in E. This makes sense since any two quasi-continuous representatives of an element (p of H1(D) are equal quasi-everywhere. For any (p e HQ(D] and t € R, the set {x G D : (f>(x) > t} is quasi-open. Moreover, the subspace HQ(^D] introduced in section 2 can now be characterized by the capacity. Corollary 2. Under the assumptions of Lemma 7.2,
The characterization of jF/Q (£);£)) requires the notion of capacity in a very essential way. It cannot be obtained by saying that the function and its derivatives are zero almost everywhere in D\Q. Recall the definition of the other extension #o(0; D} of Hl(£l) to a measurable set D containing 0 (cf. Chapter 3, section 2.5, Theorem 2.9, identity (2.29)):
By definition HQ(Q-,D) C H*(ft;D), but the two spaces are generally not equal, as can be seen from the following example. Denote by Br, r > 0, the open ball of radius r > 0 in R2. Define O = B2\dBi and D = B$. The circular crack dB\ in Jl has zero measure but nonzero capacity. Since dB\ has zero measure, H^(^l\D} contains functions i\) € H^B^} whose restriction to B2 are not zero on the circle dBi and hence do not belong to HQ (O). Yet for Lipschitzian domains 0 the two spaces are indeed equal. The following terminology is due to Rauch and Taylor [1]. Definition 7.2. il is said to be stable with respect to 7.3
Transport of Sets of Zero Capacity
Any Lipschitz continuous transformation of RN which has a Lipschitz continuous inverse transports sets of zero capacity onto sets of zero capacity. Lemma 7.3. Let D be an open subset of RN and T an invertible transformation of D such that both T and T~l are Lipschitz continuous. For any E C D
7. Elements of Capacity Theory
275
Proof, (a) E = K, K compact in D. Observe that, in the definition of the capacity, it is not necessary to choose the functions (p in C^(D}. They can be chosen in a larger space as long as
So the capacity is also given by
But the space 8 = HQ(D] fi C(D) is stable under the action of T:
Consider the matrix
By assumption on T, the elements of the matrix A belong to L°° (D) and
Define E(K) = & e E : v? > 1 on K}. For any (p e E(T(K}}, (p o T 6 £(#) and
Let K be a compact subset such that capD(K) = 0. For each e > 0 there exists (f> G E(K) such that
and we can repeat the proof with T""1 in place of T. (b) E = G, G open. Let
Therefore cap D (X) = 0, which implies that capD(T(K)) — 0 and hence
(c) General case. Let
276
Chapter 6. Optimization of Shape Functions
Therefore for all e > 0, there exists G D E open such that
since T(E] C T(G] is open. By definition of capD(G) we have
Finally, for all £ > 0, capD(T(E)) < ae and hence capD(T(£)) = 0.
8
Continuity under Capacity Constraints
In order to preserve the homogeneous Dirichlet boundary condition for u on the boundary of the limit domain 17 in section 4, it is necessary to restrict our attention to smaller families of open domains. We have to be careful since the relative capacity of the complement of the domains near the boundary must not vanish. In order to handle this point, we introduce the following concepts and terminology related to the local capacity of the complement near the boundary points. The following definition is due to Heinonen, Kilpelainen, and Martio [1]. Definition 8.1. For r > 0 and a compact K C R N , the capacity condenser of K in the ball B(x,r) is defined as The reader is referred to Hedberg [1] and Bucur and Zolesio [5] for detailed properties. Definition 8.2. (i) Given r > 0, c > 0, and an open set, fi is said to satisfy the (r, c)-capacity density condition if
(ii) For 0 < r < 1 define the following family of open subsets of D:
Remark 8.1. There are different definitions of the capacity condenser. The definition used here is given in Hedberg [1]. The following slightly different definition is given by Heinonen, Kilpelainen, and Martio [1]:
8. Continuity under Capacity Constraints
277
For a given compact set K we generally have
but the following two conditions are equivalent:
and
and so from the Oc^r families viewpoint, the two definitions are equivalent.
Definition 8.3. Let re be a point on d£l. The set CO is said to be thick at x if the complement of £1 locally has "enough" capacity, that is, if
Remark 8.2. If 0 satisfies the (r, c)-capacity density condition, the complement is thick in any point of the boundary. Recall Theorem 4.1 and the notation un for the solution of equation (4.2) in HQ(Q;D) and u for the weak limit in HQ(D) which satisfies equation (4.6). The following theorem gives the main continuity result. Theorem 8.1. Assume that D is a bounded open nonempty subset o/R N and that D is of class C2 for N > 3. Assume that A is a matrix function that satisfies the conditions (1.3) and that the elements of A belong to Cl(D}. Let {Hn} be a sequence in Oc^r(D) which converges in the Hc-topology to an open set 17. Then un —> u in HQ(D)-strong and MQ = u Q e //o(Q) is the solution of equation (1.2). Proof. When the dimension TV of the space is equal to 1 or 2, H1(D) C C(D) and no approximation of / is necessary. When N > 3 we first prove the result for / G HS(D), s > N/2 — 2 (since for D of class C2 the corresponding solution will belong to HS+2(D] C C(D}} and then, by approximation of /, we prove it for ftH-\D}. (i) First consider / e HS(D], s > N/2 - 2 for TV > 3 (s = -I for N = 1 or 2). The proof will make use of the following results from Heinonen, Kilpelainen, and Martio [I]. Lemma 8.1. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let v be an A-harmonic function, that is, the weak solution of equation (1.2) in the open set 0 for f — 0. Then v can be redefined on a set of zero measure, so that it becomes continuous in 0.
278
Chapter 6. Optimization of Shape Functions
Note that the continuous representative from Lemma 8.1 is in fact a quasicontinuous #0(0) representative. Indeed, let v\ = v almost everywhere and v\ be continuous on 0. We want to show that v\ is a quasi-continuous representative of v. There exists a quasi-continuous representative v% of v, which is equal to v almost everywhere. So v\ is continuous, v% is quasi-continuous, and v\ = i>2 almost everywhere. Using Heinonen, Kilpelainen, and Martio [1, Thm. 4.12], we get v\ = v% quasi- everywhere. Lemma 8.2. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let Q belong to Oc,r(D}. If 9 G Hl(Sl] fl C(£l] and if h is an A-harmonic function in 0 such that h — 0 G HQ(£I), then
Note that the fact that £1 belongs to OCjr(D) involves the notion of thickness in any point of its boundary, which necessarily occurs in the proof of the lemma (cf. Definition 8.3 and Hedberg [1]). Returning to Theorem 8.1, it will be sufficient to prove the continuity for a subsequence of {On}. By Theorem 4.1 there exists a subsequence of {On}, still indexed by n, such that un weakly converges to u in HQ(D] and u satisfies equation (4.6) in 0. We now prove that under our assumptions, WQ = u\fi G HQ(£I), which implies that u — en(wn). For that purpose we use Lemma 7.2 which says that it is sufficient to prove that u = 0 quasi-everywhere in D\Q for some quasi-continuous representative u. From Theorem 4.1 we already know that u = 0 quasi-everywhere in D\£l. So it remains to show that u = 0 quasi-every where in u in HQ(D}. Because of the strong convergence of {i/)n} to u in HQ(D), we have for a subsequence of {^n}, still indexed by n. Let GQ be the set of zero capacity on which {i^n(x)} does not converge to u(x). Given x G D\(£t U GO), we prove that, for all e > 0, u(x)\ < e. We have
There exists Ne>x > 0 such that for all n > N£jX,
It remains to show that there exists N' such that for all n > A7"', un(x}\ < e/2 implies i/Jn(x)\ < e/2. Denote by UD the solution of (1.2) in D. By assumption
8. Continuity under Capacity Constraints
279
on D and /, the solution up is continuous in D. Subtracting the corresponding equations, we obtain
Define
Therefore, the restriction hn of hn to $ln is .4-harmonic in Qn and, from Lemma 8.1, continuous in f£ n . Moreover, we have the continuity of u$in in the closure f) n , and utin is zero on the boundary. To show that, we use Lemma 8.2 with hn and 9n. By definition, hn — 9n = — u^ n belongs to #o(D n ). From the continuity of UD we obtain that the continuous extension hn of hn is equal to up on 90n. Hence the extension un of u^ n to the boundary is zero. Using Heinonen, Kilpelainen, and Martio [1, Thm. 6.44], we obtain that, if hn is 5-H61derian on d£ln, then there exists is (s + 2 - AT/2)-H61derian on D (with s + 2 - aV/2 > 0), with a constant M, because of the assumption on / and D. Finally, we get
So there exists 61 = 6i(N,/3/a,c) and
such that
By a simple argument we obtain that this inequality holds in D, and hence there exists 62, Si < 62 < 1, such that for all x, y 6 D we have
Choose R > 0 such that M^R5'2 < s/2. Because of the £Tc-convergence of fi n to fi there exists an integer HR > 0 such that for all n > HR we have (D\Q n ) n £?(x, ^) ^ 0. For x n e (D\fi n ) n S(x, /?),
because w n (x n ) = 0. So
280
Chapter 6. Optimization of Shape Functions
Finally we obtain \u(x)\ < E. Because £ was arbitrary, we have u(x) = 0 quasieverywhere on D\0, which implies that u G HQ (fi; D] and u$i = U|Q 6 #0(17). The strong convergence of {un} to u now follows from Theorem 4.1 and u = e^(u^). (ii) In the next step we prove that the continuity result is preserved for / G H~l(D}. The main idea is to use the continuous dependence of the solution WQJ G HQ(£I\D} with respect to /, which is uniform in 0. Indeed, let 0 C D and /,g G H~l(D}. Then, by a simple subtraction of the equations, we get
So, let / G H~1(D) and f£ — f * p£ for some mollifier p£. Letting e —» 0 we have II/£ — f\\H~1(D) -* 0- Let {On} C OCjr(D) be a sequence that converges in the #c-topology to an open set 0. Then we have
because f£ G HS(D] and, from the previous considerations,
uniformly in (ln. Given 6 > 0, we get
Choose e sufficiently small such that ||/£ — f\\H~1(D) < ^/4j and for each E choose n£js > 0 such that
As S was arbitrary, the proof is complete. We can now recover the result of Sverak [2] in dimension N = 2. Theorem 8.2. Let the assumptions of Theorem 8.1 on the matrix function A be satisfied. Let N = 2 and I > 0 be a positive integer. Define the set
where # denotes the number of connected components. Then the set Oi is compact in the Hc-topology and the map
is continuous.
9. Flat Cone Condition and the Compact Family Oc>r(D}
281
A proof of the result of Sverak [2] is given in Bucur [1] as a consequence of Theorem 8.1. The main idea of the proof is to consider / > 0 (because of the decomposition of / = /+ — / ~ ) and a sequence {£ln} C Oi such that On converges in the #c-topology to an open set Q, and u$in —x u in HQ(D}. He constructs extensions £7+ of the domains £ln with the following properties:
where c and r are suitable constants. In fact it can be proved that 0+ satisfy this capacity density condition, because in an open connected set any two points are linked by a continuous curve that lies in the set, and in the bidimensional case a curve has a positive capacity. From the previous theorem we get u^+ —^ UQ+ . We easily obtain that M^+ — u^ > u > 0, which will imply that u = 0 quasi-everywhere on CS7, and this concludes the proof. For details, see Bucur [1].
9
Flat Cone Condition and the Compact Family Oc,r(D)
In this section we introduce families of domains which have a simpler geometric description and satisfy the capacity density condition for some constants c and r. For a weaker result in that direction see also Bucur and Zolesio [1]. We next prove that the set Oc^r(D} is compact in the Hc-topo\ogy. From this we prove existence of optimal solutions for several shape functions which continuously depend on the solution of the state equation.
Definition 9.1.
Let x e R N , 0 < uj < 7T/2, A > 0, and i/, d e R N such that z/| = \d = I. The flat cone is denned as
where Cx(X,uj,d)
is the cone defined in Notation 6.1 of Chapter 2 and
The cone CXV(X, u;, d} is said to be flat since it is contained in the hyperplane H x(v], and u;, A, and d are, respectively, the aperture, the height, and the orientation of the axis of symmetry of the cone. So its TV-dimensional Lebesgue measure is zero.
Definition 9.2 (flat cone condition). Let 0 < uj < 7T/2 and A > 0 be real numbers. (i) An open set !T£ in D, is said to satisfy the (a;, X)-flat cone condition (or simply (a;, A)-f.c.c.) if, for each x E 0 and r > 0 such that
Proof. This readily follows with r = A and for each x G d£l
and the properties of the capacity with respect to translation and similarity. Theorem 9.2. The family O(u,\,D) is compact in the Hc -topology. Proof. It is sufficient to prove that O(u>, A, D) is closed. For that, consider a sequence {£7n} C O(u,\,D) and n n -^fi. We shall prove that ft G 0(u;,A,D). Given a: G dfi, there exists a sequence of points, xn G £7n such that #n —>• x. As Qn G C?(u;, A,D), there exists a flat cone CXnvn(\u,dn} C C0n. It is easy to see that there exists subsequences {dnk} and {vnk} of {rfn} and {Vn}, respectively, such that dnk —> d and z/nfe —* ZA Because of the properties of the H°-topology we obtain that CXv(X,u),d) C CH. Finally O(ui,X,D) is closed and hence compact. Corollary 3. Given a sequence {£ln} of open sets in O(uj,\,D}, then
Theorem 9.3. The family Oc^r(D] of Definition 8.2 is compact. Proof. It is sufficient to prove that
ZJ-C
Pick a sequence {f2n} C Gc,r(D) such that O n —>£l. Define
Let x G d£l. We now prove that the capacity density condition for 17 is satisfied in the point x. Because of the #c-convergence,
Given e > 0 pick n e /2 and xn G d$ln such that x — xn\ < £/2. Denote by r the translation by the vector xn — x. As rB(x,r] = B(xn,r), and the capacity condenser is invariant under the translation of the two arguments,
Because Kn C ^e/2 and \x — xn\ < £/2 we have rKe D Kn. Then, from the monotonicity in the first argument, we get
9. Flat Cone Condition and the Compact Family Oc,r(D]
283
Using the capacity density condition for Qn we have ca
Ps(x,2r 0 )(^e nB(x,r 0 )) > c cap B ( X i 2 r ( ) ) (B(x, r 0 )).
Letting e —» 0, and using the continuity of the capacity for decreasing sequences of compact sets and the fact that
we get the capacity density condition for 17 in x. Hence 17 £ Oc,r(D). A first example of extremal domain follows directly from Theorem 9.3. Theorem 9.4. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Further assume that E is a nonempty open subset of D. Further assume that A satisfies conditions (1.3). Then the maximization problem
has solutions. In some sense the maximizing solution is an OC)T.(D)-approximation of the set E. The result follows from the upper semicontinuity of the map 17 i—>• AA(17) in the ./f c-topology and the fact that the family of admissible domains is compact (Theorem 2.3 in Chapter 4 and Theorem 9.3) for the Hausdorff complementary topology. We now give the following general existence theorem. Theorem 9.5. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. Let un be the solution of (1.2) for 17 in Oc^r(D}. If h is continuously defined from HQ(D] into R, then J(17) = ^(CQ(UQ)) is continuously defined from OCjr(D] into R and reaches its extremal values on that set. Example 9.1. Consider the following shape function for which we can get the existence of minimizing domains even if the assumptions of Theorem 9.5 on h are not satisfied. Given a > 0,
where u^ is the solution of equation (1.2) and z e I/ 2 (D;R N ). From Theorem 2.1 u — eti(un) and VUQ are zero almost everywhere in D\f2, and the function can be rewritten as
The last term is not of the form h(u$i) for some h, but simply of the form h(£l). It turns out that J is not continuous but only lower semicontinuous for the Hctopology. Hence it can be minimized.
284
Chapter 6. Optimization of Shape Functions More precisely, we need the following result.
Lemma 9.1. Let IJL be a positive finite measure on D. The map 17 >—>• //(17) is lower semicontinuous with respect to the Hc-topology. Proof. Let {17n} be a sequence of open sets in £?, Kn = Z)\17n, K — D\17, and rrC
assume that 17n —> 17. Given £ > 0, we have K£ D _K"n for all n > n£. Then for all n > n£, n(K£] > (jL(Kn} and
But K£ is a monotonically decreasing sequence and then (JL(K£} —>• fJ-(K) as e —> 0, and the result follows. The minimization problem of Example 9.1 can now be formulated as follows on D. Let D be a bounded open nonempty domain in R N , / an element of H~1(D), and B a ball containing D. Let // be a positive measure on D and a a positive constant such that 0 < a < /t/(D) < +00. Let un be the solution of equation (1.2), and u = en(wn), its extension by zero in HQ(D}. For J(17) defined in (9.2) consider the following problem:
Theorem 9.6. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisfied. For any constants a > 0, c > 0 and r, 0 < r < 1, (9.4) has at least one solution. Proof. It is sufficient to notice that the first term in (9.3) is continuous under the assumptions of Theorem 9.5. The second term is lower semicontinuous from Lemma 9.1 (with the measure of density \z\2). To complete the proof, recall that, if 17n C D and 17n—>17, then 17 C D. As a > 0, a minimizing sequence cannot converge to 0. In fact, for any admissible domain 17o and any optimal solution WQ O , rrc
10
Examples with a Constraint on the Gradient
Let D be a fixed bounded smooth open domain in R N , / in H~l(D), and g in L2(D). For any open subset 17 of D, let u$i be the solution of the Dirichlet problem (1.2) in #0(17). Given a > 0, M > 0, and an open subset E of D, consider the following minimization problem:
10. Examples with a Constraint on the Gradient
285
It is understood that if |Vn| is not in L°°(D), then the esssup|Vu| is +00. In a first step it is easy to check that the problem
has minimizing solutions. Let {On} be a minimizing sequence. By assumption there exists £1 and a subsequence, still indexed by n, such that On —>• 0 in the #c-topology and Then |Vu n | converges in L2(D] to |Vw from the previous sections. For any (p £ L?+(D] we have
and then, in the limit, we get the same inequality with u. Then, the function being lower semicontinuous with respect to that topology, we get the existence. Lemma 10.1. For any f £ H~l(D) and any r > 0 and c> 0, problem (10.2) has solutions in Oc^r(D} In fact, the presence of the constraint on the gradient of u is helpful to get the continuity of the application 0 i—> u = e^(wn), and we directly get the existence of solutions to the original problem (10.1). In the proof of Theorem 8.1 we only needed the equicontinuity of the family of solutions {un = esin(unn}} and the fact that if x £ CO then u(x] = 0 for a quasi-continuous representative. If 0 6 (9C]T.(D), those two assumptions are readily satisfied. Notice that the boundedness of the gradients in (10.1) implies the equicontinuity of any minimizing sequence {un}. So, in order to obtain a continuity result with constraints on the boundedness of the gradient, we only have to notice that if u £ Hg(0), esssup|Vu| < M. Then u £ Wli°°(£l) and u is Lipschitzian with the constant M, or more exactly, there exists a Lipschitzian function almost everywhere equal to u, which is also a quasi-continuous representative of u. To obtain that u(x) — 0 in any point of the complement of 0, we have to introduce the following capacity constraints on 0: we require that 0 be capacity extended (see Bucur [1]), i.e., that 0 = O*, where
Indeed, let u be continuous on D, u = 0 quasi-everywhere on CO, and 0 = 0*. Then MX £ CO, Ve > 0 we have capD(B(x, ex] n CO) > 0 and there exists a point x£ in B(X,EX) Pi CO where u(xe] — 0. By continuity we get u(x) = 0. We recall from Bucur [1] the main properties of the capacity extension. Proposition 10.1. For any open 0 C D, the set 0* is open,
286
Chapter 6. Optimization of Shape Functions
As cap£>(17*\f2) = 0 we get e.£i(u$i) — e^(w^*) and any minimizing sequence can only be made up of capacity-extended domains. Theorem 10.1. Given /, g G L2(D), a > 0, M > 0, problem (10.1) has minimizing solutions. Proof. From the boundedness of the gradient we obtain that {w^n } are uniformly Lipschitz continuous, and we can apply the same arguments as in Theorem 8.1, avoiding the capacity conditions. Consider the penalized version of problem (10.1). Given M > 0 and (3 > 0, define
and consider the existence of solutions to the following problem: Theorem 10.2. Given f and g in L2(D), a > 0, j3 > 0, M > 0, problem (10.5) has minimizing solutions. Proof. Let {On} be a minimizing sequence for the function J^, fin = Q*. We have esssup|VunJ < max{M,/3~ 1 J(^i)} = M' for all n > 1. Then, from the previous considerations we have the strong convergence in HQ(D] of un = e^ n (wo n ) to u = en(wn), for some subsequence {Qn} that converges to f2 in the #c-topology. Hence O is a minimizing domain. To complete the proof note that the map
is not lower semicontinuous from the #c-topology to R. Nevertheless, if {fin} is a minimizing sequence for problem (10.5), the required property is satisfied. We can assume that the sequence is chosen such that
For any e > 0 there exists n£ > 0 such that for n > n e , supD ess | Vwjin | < c + e. Because of the HQ (Z?)-strong convergence of un to w, we get
Remark 10.1. We can change the L°°-norm for a differentiate one; that is, we need By the continuous inclusion W1'P(D) C WE'°°(D), e small, the result still holds.
Chapter 7
^ansformations vers s I Flows of Veloci ti
1
Introduction
In the previous chapters we have constructed examples of metric spaces of sets, which, of course, are not topological vector spaces. Studying continuity and semidifferentiability of a function denned on such spaces is analogous to studying continuity and semidifferentiability on a manifold. The two notions can be associated with the behavior of the function along continuous one-dimensional curves or along flows generated by the solutions of a differential system. Such flows are easy to generate not only in the Euclidean space but also in smooth submanifolds where line paths between two points can no longer be used. The generic result of this chapter is that continuity of a shape function with respect to the Courant metric is equivalent to its continuity along the flows of velocity vector fields in the class of transformations associated with the Courant metric. In practice this condition is much easier to satisfy on specific examples. In the next chapter flows of velocities will be adopted as the natural framework for defining shape semiderivatives. In section 2 we motivate and adapt constructions and definitions of Gateaux and Hadamard semiderivatives in topological vector spaces to shape functions defined on shape spaces. The analogue of the Gateaux semiderivative for sets is obtained by the method of perturbation of the identity operator, while the analogue of the Hadamard semiderivative comes from the velocity (speed] method. In section 3 the velocity and transformation viewpoints will be emphasized through a series of examples of commonly used families of transformations of sets. They include Cfc-domains, Cartesian graphs, polar coordinates, and level sets. In section 4 we establish the equivalence between deformations obtained by a family of transformations and deformations obtained by the flow of a velocity field. Section 4.1 gives the equivalence under relatively general conditions. Section 4.2 shows that Lipschitzian perturbations of the identity operator can be generated by the flow of a nonautonomous velocity field. In section 4.3 the conditions of section 4.2 are sharpened for the special families of velocity fields in Co(R N ,R N ), C fc (RN, R N ), and CM(R~N, R N ). The constrained case where the family of domains 287
288
Chapter 7. Transformations versus Flows of Velocities
are subsets of a fixed hold-all or universe is studied in section 5. In both sections 4 and 5 we show that under appropriate conditions, starting from a family of transformations is equivalent to starting from a family of velocity fields. This is a key result, which bridges the gap between the two points of view. Section 6 contains the central generic result of this chapter: the equivalence of the continuity of a shape function with respect to the Courant metric and its continuity alon^the flows of all velocity fields for the families Cj(R N , R N ), Ck(R™, R N ), and C' fc ' 1 (R N ,R N ) associated with the metric. This result is of both intrinsic and practical interest since it is generally easier to check the continuity along flows than with respect to the metric. In view of this equivalence the subsequent developments of this book will be based on the velocity (speed) method.
2
Shape Functions and Choice of Shape Derivatives
Recall that for a real function / : E —> R defined on a topological vector space E, the Gateaux semiderivative at x G E in the direction v G E is given as the limit (when it exists)
and the Hadamard semiderivative as
The analogue of the Gateaux semiderivative for shape functions will be obtained by the method of perturbation of the identity operator, while the analogue of the Hadamard semiderivative will come from the velocity (speed) method. The Hadamard notion of semiderivative is important in situations where the shape is parametrized and the chain rule is necessary. These basic notions and constructions will now be formally extended to deal with shape functions. We first give the definition of a shape function. Definition 2.1. Given a nonempty subset D of R N , consider the set P(D] — {ft : £1 C D} of subsets of D. The set D will be referred to as the underlying hold-all or universe. A shape function is a map
from some admissible family A of domains in P(D] into a topological space E such that for any homeomorphism T of D such that T(Q) = ft, J(ft) - J(T(ft)) for all elements 0 € A. This universe D can represent some physical or mechanical constraint, a submanifold of R N , or some mathematical constraint. In many instances it can be
2. Shape Functions and Choice of Shape Derivatives
289
chosen large and as smooth as necessary for the analysis. In the unconstrained case D is equal to R N . Consider a real-valued shape function
defined over a family A in P(R N ). All the spaces of domains considered in the previous chapters are nonlinear and nonconvex and their elements are equivalence classes of domains or transformations. So it is important to a priori specify the equivalence class under consideration and make sure that the value of the shape function J(fi) is well denned: it has the same value for all elements in the equivalence class. Even if A does not generally have a vector space structure, it is possible to consider differential quotients and their limit along one-dimensional paths around ft. Consider perturbations of the identity operator
for some family of vector fields 0 : RN —> R N . The semiderivative 0 goes to Zero , where
When 9 is restricted to some topological vector space, the continuity and the linearity of the map
can be studied, and the results are analogous to the ones in Banach spaces. Of course this approach is limited by the fact that perturbations occur along lines between 1 + 9 and the identity operator /. It is generally not applicable when D is a smooth submanifold of RN with nonzero curvature. This first approach to the definition of a shape semiderivative is not completely satisfactory since the perturbations of the identity are nonlocal transformations: the velocity field dx(t)/dt = 9(X) at the point x ( t ) — Tt(X) depends on the point X instead of x(i). In general, this will not affect first-order semiderivatives of J, but will introduce an "acceleration term" in the definition of second-order semiderivatives in Chapter 8. This can be easily fixed, and a more natural approach followed. Go back to our initial choice of deformation in (2.3) and consider t > 0 as an artificial time. Rewrite expression (2.3) as a difference
290
Chapter 7. Transformations versus Flows of Velocities
Figure 7.1. Transport o/O by the velocity field V. and in differential form
In the limit this yields a new local deformation Tt : RN —>• R N ,
defined by the solution x(t] of the differential equation
This is the basis of the velocity (speed) method. Here the velocity of the point x(t) at time t is equal to the velocity field 6 evaluated at x(t): it is now a local deformation. The construction readily extends to transformations
defined by the flow of the differential equation
for nonautonomous velocity fields V(t)(x) = V(t,x) (cf. Figure 7.1). The choice of the terminology "velocity" to describe this method is accurate but may become ambiguous in problems where the variables involved are themselves "physical velocities": this situation is commonly encountered in continuum mechanics and material sciences. In such cases it may be useful to distinguish between the "artificial velocity" and the "physical velocity." This is at the origin of the terminology speed method which has often been used in the literature. The latter terminology is convenient, but not as accurate and descriptive as velocity method. We shall keep both terminologies and use the one that is most suitable in the context of the problem at hand.
2. Shape Functions and Choice of Shape Derivatives
291
The shape semiderivative of J at 0 in the direction V will be defined as
(when the limit exists), where
and Tt is the transformation of RN defined by (2.10)-(2.11). In this book we have chosen to give all the basic definitions, constructions, and theorems within the context of the velocity method. This choice is motivated by the fact that most results based on perturbations of the identity or asymptotic developments can be readily recovered by direct application of the velocity method with nonautonomous velocity fields V(t,x). To support this assertion we now construct the time-dependent velocity field V(£,x) associated with the perturbation of the identity
The field V is to be chosen such that the function x in (2.14) is the solution of the differential equation
It is readily seen that the appropriate choice is V(t) = 9 o T^1 or
Under appropriate continuity and differentiability assumptions:
where D6(x] is the Jacobian matrix of 9 at the point x. In compact notation,
This last computation shows that at time 0, the points of the domain £7 are simultaneously affected by the velocity field V(0) = 9 and the acceleration field V"(0) — —[D9]9. Under suitable assumptions the two methods will produce the same first-order semiderivative. However, second-order semiderivatives will differ by an acceleration term which will appear in the expression obtained by the method of perturbation of the identity.
292
3
Chapter 7. Transformations versus Flows of Velocities
Families of Transformations of Domains
In section 2 we have given a formal presentation of the main steps and options leading to the definition of semiderivatives of a shape function. We have emphasized perturbations of the identity operator and flows of velocity fields. Before proceeding with a more abstract treatment, we present several examples of definitions of shape semiderivatives that can be found in the literature. We consider special classes of domains (C°°, C fc , Lipschitzian), Cartesian graphs, polar coordinates, and level sets that provide classical examples of parametrized and/or constrained deformations. In each case we construct the associated underlying family (not necessarily unique) of transformations {Tt : Q 0, consider the following perturbation Tt of T along the normal field n:
We claim that for r sufficiently small and alH, 0 < t < r, the set Tt is the boundary of a C^-domain £lt by constructing a transformation Tt of RN which maps fi onto fit and F onto Tt. First construct an extension TV e D(R N , R N ) of the normal field n on F. Define
By construction
m ^ 0 on F, and there exists a neighborhood U\ of F contained in UQ where m ^ 0 since m is at least Cl. Now construct a function r0 in V(Ui), 0 < r 0 (x) < 1, and a neighborhood V of F such that Define the vector field
Hence TV belongs to £>(R N ,R N ) since supp/V C V is compact. Moreover,
For each j, po hj G C°°(BQ) and the extensions
belong, respectively, to C°°(B} and C°°(Uj). Then
is an extension of p from F to RN with compact support since supp r3 cUj CU.
294
Chapter 7. Transformations versus Flows of Velocities Define the following transformation of R N :
By construction, pN is uniformly Lipschitzian in R N , and by Theorem 4.2 there exists 0 < r such that Tt is bijective and bicontinuous from RN onto itself. As a result from Dugundji [1] for 0 < t < T,
Since the domain 0^ is specified by its boundary I\, it only depends on p and not on its extension p. The special transformation Tt introduced here is of class C°°, that is, Tt £ C°°(RN, R N ), and \ [Tt - I] is proportional to the normal field n on F, but it is not proportional to the normal nt on Tt for t > 0. In other words, at t = 0 the deformation is along n, but at t > 0 the deformation is generally not along nt. If J(fl) is a real-valued shape function defined on C^-domains in .D, the semiderivative (if it exists) is defined as follows: for all p £ C°°(F)
It turns out that this limit only depends on p and not on its extension p.
3.2
Cfe-Domains
When J7 is a domain of class Ck with boundary F, the normal field n belongs to 6 yfc ~ 1 (r,R N ). Therefore, choosing deformations along the normal would yield transformations {Tt} mapping Cfc-domains 0 onto Cfc-1-domains ftt = Tt(0). The obvious way to deal with Cfe-domains is to relax the constraint that the perturbation pN be carried by the normal. Choose vector fields 0 in £> fc (R N ,R N ) and consider the family of transformations
This is a generalization of the family of transformations (3.8) in section 3.1 from pN to 0. For k > 1, G is again uniformly Lipschitzian in R N , and by Theorem 4.2 there exists r > Q such that Tt is bijective and bicontinuous from RN onto itself. Thus for 0 < t < T,
A more restrictive approach to get around the lack of sufficient smoothness of the normal n to F would be to introduce a transverse field p on F such that
Given p and p £ C fc (F) define for t > 0
3. Families of Transformations of Domains
295
Choosing Cfc-extensions p and p of p and p we can go back to the case where
For any 6 G D f c (R N ,R N ) the semiderivative is denned as
3.3
Cartesian Graphs
In many applications it is convenient to work with domains Q which are the hypograph of some positive function 7 in Cartesian coordinates. Such domains are typically of the form
where U is a connected open set in R and 7 G C(U', R+) is a positive function. Many free boundary and contact problems are formulated over such domains. Usually the domains Q (and hence the functions 7) will be constrained. The function 7 can be specified on U\U or not. In some examples the derivative of 7 could also be specified, that is, $7/dv = g on dU, where U is smooth and v is the outward unit normal field along dU. When 7 (resp., d^/dv] is specified along dU, the directions of deformation /z are chosen in C(U; R + ) such that
For small t > 0 and each such p, define the perturbed domain
and the obvious family of transformations
which can be extended to a neighborhood D of £7 containing the perturbed domains ^t, 0 < t < ti, for some small t\ > 0. In general, D will be such that D — U x [0, L] for some L > 0. This construction is also appropriate for domains that are Lipschitzian or of class Ck (1 < k < oo), depending on whether 7 is a Lipschitzian or a Cfc-function. Again the transformation Tt is equal to / + tO, where
296
Chapter 7. Transformations versus Flows of Velocities
But of course this 0 is not the only choice for which £lt = 7^(0). For instance, let A : R+ —> R+ be any smooth increasing function such that A(0) = 0 and A(l) = 1. Then we could consider the transformations
This example illustrates the following general principle: the transformation Tt of D such that J7t = Tt(J7) is not unique. This means that, at least for smooth domains, only the trace Tt rt on r$ is important, while the displacement of the inner points does not contribute to the definition of f^. Nevertheless this statement is to be interpreted with caution. Here we implicitly assume that the objective function J(17) and the associated constraints are only a function of the shape of fi. However, in some problems involving singularities at inner points of fi (e.g., when the solution y(0) of the state equation has a singularity or when some constraints on the domain are active), the situation might require a finer analysis. One such example is the internal displacement of the interior nodes of a triangularization T^ when the solution y(£l) of a partial differential equation is approximated by a piecewise polynomial solution over the triangularized domain fi^ in the finite element method. Such a displacement does not change the shape of f^, but it does change the solution yh of the problem. When the displacement of the interior nodes is a priori parametrized by the boundary nodes the solution yh will only depend on the position of the boundary nodes, but the interior nodes will contribute through the choice of the specified parametrization.
3.4
Polar Coordinates
In some examples domains are star-shaped with respect to a point. Since a domain can always be translated, there is no loss of generality in assuming that this point is the origin. Then such domains 17 can be parametrized as follows:
where SN-I is the unit sphere in R N ,
and / : SN-I —* R+ is a positive continuous mapping from SN-I such that
Given any g G C(SN-I) and a sufficiently small t > 0 the perturbed domains are defined as
For example, choose t, 0 < t < ti, for some
3. Families of Transformations of Domains
297
and define the transformation Tt as
As in the previous example Tt is not unique, and for any continuous increasing function A : R + —> R+ such that A(0) — 0 and X(t) = 1 the transformation
yields the same domain fi t .
3.5
Level Sets
In sections 3.1-3.4 the perturbed domain 0^ always appears in the form £lt = Ti(Q), where Tt is a bijective transformation of RN and Tt is of the form I+t®. In some free boundary problems (e.g., plasma physics, propagation of fronts) the free boundary F is a level curve of a smooth function u defined over an open domain D. Assume that D is bounded open with smooth boundary 3D. Let u G C2(D) be a positive function on D such that
If m = max {u(x) : x G D}, then for each t in [0,ra[ the level set
is a C2-submanifold of R N in D, which is the boundary of the open set
By definition, OQ = D, for all t\ > £2, ^ti C Qt 2 , and the domains £lt converge in the Hausdorff topology to the point xu. The outward unit normal field on Tt is given by
This suggests introducing the velocity field
which is continuous everywhere but at x = xu. If V was continuous everywhere, then for each X, the trajectory x(t; X) of the differential equation
298
Chapter 7. Transformations versus Flows of Velocities
would have the property that
since formally
This means that the map X i—» Tt(X] = x(t; X) constructed from (3.33) would map the level sets
onto the level set
and eventually OQ onto £lf Unfortunately, it is easy to see that this last property fails on the function u(x) = 1 — x2 defined on the unit disk. To get around this difficulty, introduce for some arbitrarily small £, 0 < £ < ra/2, an infinitely differentiate function p£ : RN —> [0,1] such that
and the velocity
As above, define the transformation
where x(t\ X] is the solution of the differential equation
For 0 < t < m — 2e, Tt maps FQ onto F f ; for 0 < s < m — E such that s +1 < m — £, Tt maps Fs onto Tt+s. However, for s > m — £, Tt is the identity operator. As a result, for 0 < t < m — 2£
Of course s > 0 is arbitrary and we can make the construction for t's arbitrary close to m. This is an example that can be handled by the velocity (speed) method and not by a perturbation of the identity. Here the domains f^ are implicitly constrained
4. Unconstrained Families of Domains
299
to stay within the larger domain D. We shall see in section 5 how to introduce and characterize such a constraint. Another example of description by level sets is provided by the oriented distance function for some open domain £7 of class C2 with compact boundary F. We have seen in Chapter 5 that there exists h > 0 and a neighborhood
such that &Q G C2 (Uh(F)). Then for 0 < t < h the flow corresponding to the velocity field V = V&Q maps £7 and its boundary F onto
4
Unconstrained Families of Domains
In this section we study equivalences between the velocity method (cf. Zolesio [12, 8]) and methods using a family of transformations. In section 4.1 we give some general conditions to construct a family of transformations of RN from a nonautonomous velocity field. Conversely, we show how to construct a nonautonomous velocity field from a family of transformations of R N . This construction is applied to Lipschitzian perturbations of the identity in section 4.2. In section 4.3 the equivalences of section 4.1 are specialized to velocities in Cj +1 (R N ,R N ), C f c + 1 (RN,R N ), andC f c ' 1 (RN,R N ), k > 0. 4.1
Equivalence between Velocities and Transformations
Let the real number r > 0 and the map V : [0, r] x RN —> R N be given. The map V can be viewed as a (time-dependent) nonautonomous velocity field {V(t) : 0 < t < T} defined on R N :
Assume that
where V(-,rr) is the function t H^ V(£,:r). Note that V is continuous on [0,r] x R N . Hence it is uniformly continuous on [0,r] x D for any bounded open subset D of RN and
Associate with V the solution x(t; V] of the vector ordinary differential equation
300
Chapter 7. Transformations versus Flows of Velocities
and define the transformations
and the maps (whenever the inverse of Tt exists)
Notation 4.1. In what follows we shall drop the V in Ty(t,X), Tyl(t,x), and Tt(V) whenever no confusion arises. Theorem 4.1. (i) Under assumption (V) the map T specified by (4.4)-(4.6) has the following properties:
(Tl) (T2) (T3) (ii) Given a real number r > 0 and a map T : [0,r] x RN —>• R N satisfying assumptions (Tl) to (T3), the map
satisfies conditions (V), where T^1 is the inverse of X H-> T t ( X ) — T(t,X). If, in addition, T(0, •) = I, then T(-,X) is the solution of (4.4) for that V. (\\i) Given a real number r > 0 and a map T : [0, r] x R N —» RN satisfying assumptions (Tl) and (T2) and T(0, •) = /, then there exists r' > 0 such that the conclusions of part (ii) hold on [0, r'}. A more general version of this theorem for constrained domains (Theorem 5.1) will be given and proved in section 5.1. Proof, (i) Conditions (Tl) follow by standard arguments. Conditions (T2): Associate with X in RN the function
Then
4. Unconstrained Families of Domains
301
For each x G R N , the differential equation
has a unique solution in C 1 ([0,t];R N ). The solutions of (4.11) define the map
such that
In view of (4.10) and (4.11)
To obtain the other identity, consider the function
where y(-,x) is the solution of (4.11) for some arbitrary x in R N . By definition
and necessarily
Conditions (T3). The uniform Lipschitz continuity in (T3) follows from (4.12). and we only need to show that
Given t in [0,r] pick an arbitrary sequence {tn}, tn —>• t. Then for each x G R N there exists X G RN such that
from the first condition (Tl). But
By the uniform Lipschitz continuity of T^1
302
Chapter 7. Transformations versus Flows of Velocities
and the last term converges to zero as tn goes to t. (ii) The first part of condition (V) is satisfied since, for each x 6 R N and t, s in [0,r],
Thus from (T3) and (Tl) t i—>• V(t,x) is continuous at s = £, and hence for all x in R N , V(.,x) e C([0, r];R N ). The Lipschitzian property follows directly from the Lipschitzian properties (Tl) and (T3): for all x and y in R N ,
This proves that V satisfies condition (V). (iii) From (Tl) and (T2) for f ( i ) = Tt - I and t > s,
For r' = min{r, l/(2c 2 )} and 0 < t < T', c(/(f» < 1/2,
and the second condition (T3) is satisfied on [0,r']. The first one follows by the same argument as in part (i). Therefore the conclusions of part (ii) are true on [0,r'].
This equivalence theorem says that we can start from either a family of velocity fields (V(£)} on RN or a family of transformations {Tt} of RN provided that the map V, V(t,x) = V(t)(x), satisfies (V) or the map T, T(t,X) = Tt(X), satisfies (Tl) to (T3). Starting from V, the family of homeomorphisms {Tt(V}} generates the family
of perturbations of the initial domain fi. Interior (resp., boundary) points of fi are mapped onto interior (resp., boundary) points of fV This is the basis of the velocity method which will be used to define shape derivatives.
4. Unconstrained Families of Domains
4.2
303
Perturbations of the Identity
In examples it is usually possible to show that the transformation T satisfies assumptions (Tl) to (T3) and construct the corresponding velocity field V defined in (4.9). For instance, consider perturbations of the identity to the first (A = 0) or second order: for t > 0 and X G R N ,
where U and A are transformations of R N . It turns out that for Lipschitzian transformations U and A, assumptions (Tl) to (T3) are satisfied in some interval [0,r]. Theorem 4.2. Let U and A be two uniform Lipschitzian transformations o/R N : 3c> 0 such that for all X, Y e R N 7
There exists r > 0 such that the map T given by (4.14) satisfies conditions (Tl) to (T3) on [0,r]. The associated velocity V given by
satisfies condition (V) on [0, T] .
Remark 4.1. Observe that from (4.14) and (4.15)
where DU is the Jacobian matrix of U. The term V"(0) is an acceleration at t = 0 which will always be present even when A = 0, but can be eliminated by choosing A = [DU}U.
Proof, (i) By the definition of T in (4.14), t H-> T(t, X] and t^3£(t,X) = U(X] + tA(X) are continuous on [0, oo[. Moreover, for all X and Y,
and
Thus condition (Tl) is satisfied for any finite r > 0. To check condition (T2), consider for any Y e RN the mapping h(X) =f Y - [Tt(X) - X}. For any Xl and X2
304
Chapter 7. Transformations versus Flows of Velocities
For T = min{l,l/(4c)}, and any t, 0 < t < r, tc [1 + t/2] < 1/2 and h is a contraction. So for all 0 < t < r and Y £ R N , there exists a unique X G RN such that
and Tt is bijective. Therefore, (T2) is satisfied in [0,r']. The last part of the proof is the uniform Lipschitzian property of T^1. In view of (4.17), for alH, 0 < t < T, tc[I + t/2] < 1/2 and
In view of condition (T2) for all x and y,
To complete our argument we prove the continuity with respect to t for each x. Let X = T~l(x}. For any s in [0,r]
and in view of (4.18)
The continuity of T~l(x] at s = t now follows from the continuity of TS(X) at s — t. Thus condition (T3) is satisfied.
4.3
Equivalence for Special Families of Velocities
In this section we specialize Theorem 4.1 to velocities in C M (RN, R N ), CQ +I (R N , R N ), and C fc+1 (RN,R N ), k > 0. The following notation will be helpful:
whenever T^1 exists and the identities
4. Unconstrained Families of Domains
305
Recall also for a function F : R N —> RN the notation
Theorem 4.3. Let k > 0 be an integer. (i) Given T > 0 and a velocity field V such that
for some constant c > 0 independent o f t , the map T given by (4.4)-(4.6) satisfies conditions (Tl), (T2), and
for some constant c > 0 independent oft. Moreover, condition (T3) is satisfied and there exists r' > 0 such that
for some constant c independent o f t . (ii) Given r > 0 and T : [0, r] x R N —> R N satisfying conditions (4.20) and T(0, •) = I, there exists T' > 0 such that the velocity field V(i] = f ' ( t ) o T^"1 satisfies conditions (V) and (4.19) in [0,-r'j. Proof. We prove the theorem for fc = 0. The general case is obtained by induction over k. (i) By assumption on V', the conditions (V) given by (4.2) are satisfied and by Theorem 4.1 the corresponding family T satisfies conditions (Tl) to (T3). Conditions (4.20) on f . For any x and s < t
By assumption on V and Gronwall's inequality
306
Chapter 7. Transformations versus Flows of Velocities
for another constant c independent of t. Moreover,
for some other constant c' by the second condition (Tl). Again by Gronwall's inequality, there exists another constant c such that
Moreover, /'(£) = V(t) o Tt and
Finally,
and c(/'(t)) < c for some new constant c independent of t. Therefore,
Conditions (4.21) on g. Since conditions (Tl) and (T2) are satisfied there exists r' > 0 such that conditions (T3) are satisfied by Theorem 4.1 (hi). Moreover, from conditions (4.20)
4. Unconstrained Families of Domains
307
Choose a new r" = minJT', l/(2c)}. Then for 0 < t < r", c(g(t)) < let. Now
For t in [0,r"], ct < 1/2, and
The conditions (4.20) on / are satisfied for k = 0. For /e = 1 we start from the equation
and use the fact that DT^1 = [DTt]~l o T^1 in connection with the identity
(ii) From conditions (4.20) on /, the transformation T satisfies conditions (Tl). To check condition (T2) we consider two cases: k > 1 and k — 0. For k > 1 the function t \-> Df(t] = DTt - I : [0,r] -> C' fc - 1 (RN,R N ) 7V is continuous. Hence t i—>• detDTt : [0,r] —> R is continuous and detDTo — 1. So there exists r' > 0 such that Tt is invertible for all t in [0,r'] and (T2) is satisfied in [0,r']. In the case /c = 0 consider for any Y the map h(X) =Y — f(t}(X). For any X\ and X2, |/i(X2) - /i(Xi) < c(/(t)) |^2 - Xi|. But by assumption / € ^([O,^;^ 0 '^^)) and c(/(0)) = 0 since /(O) = 0. Hence there exists r' > 0 such that c(/(t)) < 1/2 for all t in [0, T'} and /i is a contraction. So for all Y in R N there exists a unique X such that
Tt is bijective, and condition (T2) is satisfied in [0,r']. By Theorem 4.1 (iii) from (Tl) and (T2), there exists another r' > 0 for which conditions (T3) on g and (V) on V(t) = /'(£) oXp 1 are also satisfied. Moreover, we have seen in the proof of part (i) that conditions (4.21) on g follow from (T2) and (4.20). Using conditions (4.20) and (4.21),
308
Chapter 7. Transformations versus Flows of Velocities
and c(V(i)} < c'. Also
Therefore, since both g and /' are continuous,
for some constant c independent of t. This proves the result for k = 0. As in part (i), for k = 1 we use the identity
and proceed in the sane way. The general case is obtained by induction over k. We now turn to the case of velocities in CQ (R N , R N ). As in Chapter 2, it will be convenient to use the notation CQ for the space I be an integer. (i) Given T > 0 and a velocity field V such that
the map T given by (4.4)-(4.6) satisfies conditions (Tl), (T2), and
Moreover, conditions (T3) is satisfied and there exists r' > 0 such that
(ii) Given r > 0 and T : [0, r] x RN —> RN satisfying conditions (4.24) and T(0, •) = /, there exists T' > 0 such that the velocity field V(t) = f ' ( t ) o T^1 satisfies conditions (V) and (4.23) on [0,r']. Proof. As in the proof of Theorem 4.3 we only prove the theorem for k = 1. The general case is obtained by induction on fc, the various identities on /, g, /', and V, and the techniques of Theorem 8.1 and Lemmas 8.2 and 8.3 in Chapter 2. (i) By the embedding Cg(R N ,R^) C C 1 (R N ,R N ) C C°' 1 (R N ,R N ), it follows from (4.23) that V G CQO^jC^R^R 1 *)) and condition (4.19) of Theorem 4.3 are satisfied. Therefore, conditions (4.20) and (4.21) of Theorem 4.3 are also satisfied in some interval [0, r'], T' > 0.
4. Unconstrained Families of Domains
309
Conditions (4.24) on f . It remains to show that f(t) and /'(£) belong to the subspace Co(R N ,R N ) of C(R N ,R N ) and to prove the appropriate properties for Df(t) and D f ' ( t ) . Recall from the proof of the previous theorems that there exists c > 0 such that
By assumption on V(0), for E > 0 there exists a compact set K such that
and there exists £, 0 < 6 < 1, such that
Proceeding in this fashion from the interval [0, 6] to the next interval [ 0 there exists a compact set K(t) such that |l/(t)(x)| < e on C/C(t). Thus by choosing the compact K't = T t ~ l ( K ( t ) ) , f'(t)(x)\ < £ on C^, and /' e C([0,r];C 0 ). In order to complete the proof, it remains to establish the same properties for Df(t) and D f ' ( i ) . The matrix Df(t) is solution of the equations
From the proof of Theorem 8.1 in Chapter 2 for each t the elements of the matrix
belong to Co since DV(t] and /(t) do. By assumption, V € (7([0,r];Co) and I/ and all its derivatives daV are uniformly continuous in [0,r] x R N . Therefore, for each e > 0 there exists 6 > 0 such that
310
Chapter 7. Transformations versus Flows of Velocities
Pick 0 < 6' < 6 such that
For each x, Df(t}(x) is the unique solution of the linear matrix equation (4.26). To show that D f ( t ) 6 (Co]N we first show that Df(t)(x) is uniformly continuous for x in R N . For any x and y
But / e C([0,r];Co) is uniformly continuous in ( t , x ) : for each s > 0 there exists 6, 0 < 6 < £/(2cr), such that
Substituting in the previous inequality for each e > 0, there exists 6 > 0 such that
Hence Df(t) is uniformly continuous in R N . Furthermore, from the equation (4.26) we have the following inequality:
since V G C([0, r];Cg). By Gronwall's inequality,
for some other constant c independent oft. Thence Df(t) E C(R N ,R N ) j V . Finally to show that Df(t) vanishes at infinity we start from the integral form of (4.26):
4. Unconstrained Families of Domains
311
By the same technique as before for /(t), it follows that the elements of Df(t) belong to CQ since both /(s) and DV(r) do. Finally for the continuity with respect to t
Again, by Gronwall's inequality, there exists another constant c such that
Therefore, Df G C([0,r]; (C0)N) and / G C([0,r];Cj). For Df we repeat the proof for /' using the identity
to get
Conditions (4.25) on g. From the remark at the beginning of part (i) of the proof, the conclusions of Theorem 4.3 are true for g, and it remains to check the remaining properties for g and Dg using the identities
By the proof of Theorem 8.1 in Chapter 2, g(t) G CQ since Df(t) and g(t) do. Therefore, g(t) G CQ. The continuity follows by the same argument as for /' and peCaO.rhCo 1 ). (ii) By assumption from conditions (4.24), conditions (Tl) are satisfied. For (T2) observe that for k > 1 the function t i-> Df(t) = DTt - I : [0,r] -» C' fc - 1 (RN,R N ) 7V is continuous. Hence t ^ det DTt : [0,r] -> R is continuous and detDTo = 1. So there exists r' > 0 such that Tt is invertible for all t in [0,r'] and (T2) is satisfied in [0,r']. Furthermore, from the proof of part (i) conditions (T3) and (4.25) on g are also satisfied in some interval [0,r'], r' > 0. Therefore, the velocity field
satisfies the conditions (V) specified by (4.2) in [0,r']. By the proof of Theorem 8.1 in Chapter 2, V(t] e CQ since f ' ( t ) and g(t) belong to CQ. By assumption, / G C 1 ([0,r];Cg)- Hence /' and all its derivatives 5a/', a\ < fc, are uniformly continuous on [0, r] x R N ; that is, given e > 0, there exists 6 > 0 such that
312
Chapter 7. Transformations versus Flows of Velocities
Similarly g € (7([0, r'];CJo) and there exists 0 < 6' < 6 such that
Therefore, for \t - s\ < 6'
and since 6' < 6
We then proceed to the first derivative of V,
and by uniform continuity of the right-hand side V 6 C([0,r'];Co). By induction on fc, we finally get V € C([0, T'];Cj). The proof of the last theorem is based on the fact that the vector functions involved are uniformly continuous. The fact that they vanish at infinity is not an essential element of the proof. Therefore, the theorem is valid with C f c (R N ,R N ) in place of C$(R N ,R N ). Theorem 4.5. Let k > I be an integer. (i) Given r > 0 and a velocity field V such that
the map T given by (4.4)-(4.6) satisfies conditions (Tl), (T2), and
Moreover, conditions (T3) are satisfied and there exists r' > 0 such that
(ii) Given r > 0 and T : [0, r] x RN —>• RN satisfying conditions (4.29) and T(0, •) = /, there exists r' > 0 such that the velocity field V(t) = f ' ( t ] o T^1 satisfies conditions (V) and (4.28) on [0,r'].
5
Constrained Families of Domains
We now turn to the case where the family of admissible domains fJ is constrained to lie in a fixed larger subset D of RN or its closure. For instance, D can be an open set or a closed submanifold of R N .
5. Constrained Families of Domains
5.1
313
Equivalence between Velocities and Transformations
Given a nonempty subset D of R N , consider a family of transformations
with the following properties:
where under assumption (T2/?), T~l is denned from the inverse of Tt as
Those three properties are the analogue for D of the same three properties obtained for R N . In fact, Theorem 4.1 extends from RN to D by adding one assumption to (V). Specifically we shall consider for r > 0 velocities
such that
where Tp(x) is the Bouligand contingent cone to D at the point x in D
and B is the unit disk in RN (cf. Aubin and Cellina [1, p. 176]). This definition is equivalent to
(cf. Aubin and Frankowska [1, pp. 121-122; 17; 21]). Note that when D is bounded inR N ,
314
Chapter 7. Transformations versus Flows of Velocities
When D is equal to R N , Tp(x) = RN for all x and condition (V2p) can be dropped. When D is equal to the boundary dA of a set A of class C1'1 in R N , dA is a C1'1submanifold of RN and
that is, at each point of dA, the velocity field is tangent to dA: it belongs to the tangent linear space of dA. The next theorem is a generalization of Theorem 4.1 from RN to an arbitrary set D which shows the equivalence between velocity and transformation viewpoints.
Theorem 5.1. (i) Let T > 0 and V be a family of velocity fields satisfying conditions (Vl£>) and (V2/)) and consider the family of transformations
where x ( - , X ) is the solution of
Then the family of transformations T satisfies conditions (Tl/p) to (T3£>). (ii) Conversely, given a family of transformations T satisfying conditions (Tl/j) to (T3.c>), the family of velocity fields
satisfies conditions (Vlp) and (V2z?). //, in addition, T(0, •) = I, then T(-, X) is the solution of (5.9) for that V. (iii) Given a real number r > 0 and a map T : [0, T] x D —> D satisfying assumptions (Tl£)) and (T2^)) and T(0, •) = /, then there exists r' > 0 such that the conclusions of part (ii) hold on [0, T'} . Remark 5.1. Assumption (V2rj} is a double viability condition. Nagumo's [1] usual viability condition
is a necessary and sufficient condition for a viable solution to (5.9):
(cf. Aubin and Cellina [1, pp. 174 and 180]). Condition (V2D):
5. Constrained Families of Domains
315
is a strict viability condition which says that Tt maps D into D and
In particular, it maps interior points onto interior points and boundary points onto boundary points (cf. Dugundji [1, pp. 87-88]).
Remark 5.2. Condition (V2£>) is a generalization to an arbitrary set D of the following condition used by Zolesio [12] in 1979: For all x in 3D,
Proof of Theorem 5.1. (i) Existence and uniqueness of viable solutions to (5.9). Apply Nagumo's [1] theorem to the augmented system on fO,r]:
that is,
where x(t) = (z 0 (t),z(t)) e R N+1 , V(x) = ( l , V ( x ) ) , and
It is easy to check that systems (5.15) and (5.16) are equivalent on [0,r] and that x(t) = (t,x(t)). The new velocity field on V C R +1 is continuous at each point x £ D by the first assumption (Vlrj} since
and for 0 < XQ , yo < r
In addition,
316
Chapter 7. Transformations versus Flows of Velocities
Moreover, V(D] is bounded and R^+1 is finite-dimensional. By using the version of Nagumo's theorem given in Aubin and Cellina [1, Thm. 3, part (b), pp. 182-183], there exists a viable solution x to (5.16) for all t > 0. In particular,
which is necessarily of the form x(t) = (£, x(i)}. Hence there exists a viable solution x,x(t} G D on [0, r], to (5.9). The uniqueness now follows from the Lipschitz condition (VID}- The Lipschitzian continuity (Tip) can be established by a standard argument. Condition (T2/?). Associate with X in D the function
Then
For each x G D, the differential equation
has a unique viable solution in C1([0, £];R N ):
since by assumption (F2/))
The proof is the same as above. The solutions of (5.19) define a Lipschitzian mapping such that
Now in view of (5.18) and (5.19)
To obtain the other identity, consider the function
where y ( - , x ) is the solution of equation (5.19). By definition,
5. Constrained Families of Domains
317
and necessarily
Condition (TS^)- The uniform Lipschitz continuity in (T3£>) follows from (5.21) and (T2^), and we need only to show that
Given t in [0,r], choose an arbitrary sequence {tn}, tn —-> t. Then for each x e D there exists X £ D such that
from (Tl£>). But
By the uniform Lipschitz continuity of T^1
and the last term converges to zero as tn goes to t. (ii) The first condition (Vl^) is satisfied since for each x £ D and t, s in [0, r]
The second condition (Vl£>) follows from (T1D) and (T3.o) and the following inequality: for all x and y in D
To check condition (V2£>), recall the definition (5.6) of Bouligand contingent cone:
Chapter 7. Transformations versus Flows of Velocities
318
where B is the unit disk in R N . We first show that
By (T2/)), Tt is bijective. So it is equivalent to show that
For simplicity we use the notation
By the definition of Tp(x(t)), we must prove that
such that x'(t] G u + eB and x(t) + hu 6 D. Choose 5,0 < 6 < a, such that
Then fix i',0< t' -t ) is
which is equivalent to proving that
or with the simplified notation
5. Constrained Families of Domains
319
We proceed exactly as in the proof of (5.22) except that we choose t' such that 0 < t - t' < 6, h = t - t', and u = -[x(t] - x(t'}]/(t - t'}. Then
and
and we get (5.23). (iii) From (T1D) and (T2 D )
For r'
and the second condition (T3) is satisfied on [0,r']. The first one follows by the same argument as in part (i). Therefore, the conclusions of part (ii) are true on [0, r']. This completes the proof of the theorem. D
5.2 Transformation of Condition (V2D) into a Linear Constraint Condition (V2^) is equivalent to
since T^(x] = TD(X}. If TD(X) were convex, then the above intersection would be a closed linear subspace of R N . This is true when D is convex. In that case TD(X) — CD(X}-, where CD(X] is Clarke tangent cone and
is a closed linear subspace of R N . This means that (V2#) reduces to
It turns out that for continuous vector fields V(t, •), the equivalence of (V2£>) and (5.26) extends to arbitrary domains D. This equivalence generally fails for discontinuous vector fields. Other equivalences might be possible between TO and some intermediary convex cone between CD and TD, but there is no evidence so far of that fact. For smooth bounded open domains Q, the two cones coincide and the condition reduces to V • n = 0, n the normal to ) and (V2c) is a direct consequence of the following lemma. Lemma 5.1. Given a vector field W € C(D; R N ), the following two conditions are equivalent:
(ii) The set Lrj(x) is closed as the intersection of two closed sets. To show that it is linear, we show that for all a 6 R and V e Lp(x), aV G LD(X), and for all V and W in LD(X), V + W G LD(X). Since ±CD(X) are cones,
By convexity of ±Cx>(z)
This completes the proof of the theorem. Proof of Lemma 5.1. Assume that (5.29) is verified. By definition CD(X) C TD(X) and (5.29) =>• (5.28). Conversely, either x is an isolated point and Tp(x) = {0} = CD(X), or there are points x ^ y G D such that y —^ x. In the latter case we know that
(cf., for instance, Aubin and Frankowska [1, Thm. 4.1.10, section 4.1.5, p. 130]). Since W is continuous in D and (5.28) is satisfied, then for each x G D
and (5.28) implies (5.29).
6. Continuity of Shape Functions
321
Remark 5.3.
Lemma 5.1 essentially says that for continuous vector fields we can relax the condition of Nagumo's [1] theorem from (V2£>) involving the Bouligand contingent cone to (V2cO involving the smaller Clarke convex tangent cone. In dimension N = 3, LD(X] is {0}, a line, a plane, or the whole space.
Notation 5.1.
In what follows, it will be convenient to introduce the following spaces and subspaces:
and for an arbitrary domain D in RN
For any integers k > 0 and m > 0 and any compact subset K of RN define the following subspaces of £:
where ~Dk(K: R N ) is the space of fc-times continuously differentiable transformations of RN with compact support in K. In all cases V^' C CK- As usual T>°°(K, R N ) will de written T>(K,RN).
6
Continuity of Shape Functions
In this section we give a characterization of the continuity of a shape function
defined on a family A in ^(R1^) (cf. Definition 2.1) with values in a Banach space B with respect to the Courant metric in terms of its continuity along the flows generated by a family of velocity fields using the equivalence Theorems 4.3, 4.4, and 4.5. Checking the continuity along flows is usually easier and more natural. We specifically consider the continuity of shape functions with respect to the Courant metric associated with the quotient spaces of transformations FQ /l(lV*)/g(n) of sectionjU in Chapter 2 corresponding to the families of velocity fields C$(R N ,R N ), C fc (RN,R N ), and C M (R*,R N )6.1
Courant Metrics and Flows of Velocities
We start with the space Cj(R N ) = Cj(R N ,R N ) used in Micheletti [1].
322
Chapter 7. Transformations versus Flows of Velocities
Theorem 6.1. Let k >l be an integer, B a Banach space, and ft a nonempty open subset o/R N . Consider a shape function J : N$i([I}} —»• B defined in a neighborhood Nfi([I}) of [I] in FQ/G(£L}. Then J is continuous at ft for the Courant metric if and only if
for all families of velocity fields {V(t) : 0 < t < T} satisfying the condition
Proof. It is sufficient to prove the theorem for a real-valued function J. The Banach space case is readily obtained by considering the new real-valued function j(T) =
\j(T(ft)) - j(ft)\.
(i) If J is (5-continuous at Q, then for all e > 0 there exists 6 > 0 such that
Condition (6.3) on V coincides with condition (4.23) of Theorem 4.4, which implies conditions (4.24) and (4.25):
But by definition of the metric 6,
and we get the convergence (6.2) of the function J(Tt(ft)) to J(ft) as t goes to zero for all V satisfying (6.3). (ii) Conversely, it is sufficient to prove that for any sequence {[Tn]} such that J(ft), then necessarily i = J(ft). The same reasoning applies to the limsup and hence the whole sequence J(Tn(ft)) converges to J(fi) and we have the continuity of J at ft. We prove that we can construct a velocity V associated with a subsequence of {Tn} verifying conditions (4.23) of Theorem 4.4 and hence conditions (6.3). By Corollary 1 of Theorem 8.2 in Chapter 2 and the same technique as in the proofs
6. Continuity of Shape Functions
323
of Lemma 8.7 and Theorem 8.3 in Chapter 2, associate with a sequence {Tn} such that I set tn = f2rn and observe that tn — tn+\ = — 2~(n+1\ Define the following Cl-interpolation in (0,1/2]: for t in [t n+ i,t n ]
where p G P3[0,1] is the polynomial of order 3 on [0,1] such that p(0) = 1 and p(l) = 0 and p^(0) = 0 = p (1) (l)Conditions on f. By definition for all t, 0 < t < 1/2, /(t) = Tt -1 e Cj(R N ). Moreover, for 0 < t < 1/2
f (t) = 8T/dt(t,.) e C 0 fe (R N ) and /(•)(*) = T(-,X) - 7 e ^((0,1/2]; R N ). By definition, /(O) = 0. For each 0 < t < 1/2 there exists n > N such that tn+i 0 there exists 5 > 0 such that
324
Chapter 7. Transformations versus Flows of Velocities
In particular there exists N > 0 such that for all n > N, tn < 5, and
and this proves the 5-continuity for the subsequence {Tn}. The case of the Courant metric associated with the space C fc (R N ) = CA:(RN, RN is a corollary to Theorem 6.1. Theorem 6.2. Let k > 1 be an integer, B a Banach space, and f2 a nonempty open subset o/R N . Consider a shape function J : Afo([/]) —>• B defined in a neighborhood Afo([/]) of [I] in ^ r/c (R N )/^(n). Then J is continuous at ft for the Courant metric if and only if
for all families of velocity fields {V(t) : 0 < t < T} satisfying the condition
The proof of the theorem for the Courant metric topology associated with the space C M (RN) = C M (RN, R N ) is similar to the proof of Theorem 6.1 with obvious changes. Theorem 6.3. Let k > 0 be an integer, 0 a nonempty open subset o/R N , and B a Banach space. Consider a shape function J : Afo([/]) —» B defined in a neighborhood Nfi([I}) of [I] in J? rfc ' 1 (R N )/^(0). Then J is continuous at fi for the Courant metric if and only if
for all families {V(t) : 0 < t < T} of velocity fields in C fc)1 (R N ,R N ) satisfying the conditions
for some constant c independent of t. Proof. As in the proof of Theorem 6.1, it is sufficient to prove the theorem for a real-valued function J. (i) If J is 5-continuous at Q, then for all £ > 0 there exists 6 > 0 such that
Under condition (6.7), from Theorem 4.3
6. Continuity of Shape Functions
325
But by definition of the metric 6 and we get the convergence (6.6) of the function J(T t (Q)) to J(0) as t goes to zero for all V satisfying (6.7). (ii) Conversely, as in the proof of Theorem 6.1, it is sufficient to prove that given any sequence {[Tn]} such that 0 there exists a subsequence such that By Theorem 9.1 (i) and the same technique as in the proofs of Lemma 8.7 and Theorem 8.3 of Chapter 2, associate with a sequence {Tn} such that 5([Tn], [/]) —» 0 a subsequence, still denoted {Tn}, such that For n > 1 set tn = 2~ n and observe that tn—tn+i = — 2~( n+1 ). Define the following (^-interpolation in (0,1/2]: for t in [t n+ i,t n ],
where p G -P3[0,1] is the polynomial of order 3 on [0,1] such that p(0) = 1 and p(l) = 0 and pW(Q) = 0 = p (1) (l)Conditions on f. By definition for all t, 0 < t < 1/2, f ( t ) =Tt-Ie Ck'l(BF). Moreover, for 0 < t < 1/2
/'(t) = dT/dt e C^HR^) and /(-)PO - T(-,X) -I £ Cl((Q, 1/2];RN). By definition, /(O) = 0. For each 0 < t < 1/2 there exists n > N such that tn+i 0 (resp., Tt is specified by
where 9 e Lip (R N ,R N ) is a uniform Lipschitzian transformation of R N :
(ii) / is said to be shape continuous at 0 with respect to a class V of families of velocity fields if it is shape continuous at 0 in the direction V for all V in the class V satisfying conditions (V) given by (4.2). (iii) / is said to be shape continuous at 0 with respect to a family 0 of transformations in Lip (R N ,R N ) if it is shape continuous at 0 in the direction 9 for all 0 e 6.
This page intentionally left blank
Chapter 8
jijiape Derivatives and calculus, and Tangential Differential Calculus
1
Introduction
Chapter 7 has studied the equivalence between the two points of view of transformations and velocities. Equivalent characterizations of shape continuity by the Courant metric and continuity along flows of velocities have been established. Since the velocity approach readily extends to the constrained case, and in particular to submanifolds of R N , this chapter adopts the velocity framework to study shape semiderivatives. In section 2 we give a self-contained review of semiderivatives and derivatives in vector spaces in order to prepare the ground for shape derivatives. Section 3 gives the definitions and the main properties of first-order semiderivatives and derivatives of shape functions. Section 3.2 specializes the definitions and results of section 3 to the generic shape spaces endowed with the Courant metric. Finally, a canonical definition of the shape gradient is given along with the main structure theorem in section 3.3. Before going to second-order derivatives, the main elements of the shape calculus are introduced in section 4 and the basic formulae for domain and boundary integrals are given in sections 4.1 and 4.2. Their application is illustrated in a series of examples in section 4.3. The final expressions of the gradients in the examples of section 4 always lead to a domain and a boundary expression. The boundary expression contains fundamental properties of the gradients and the natural way to untangle some of the resulting terms is to use the tangential calculus. Section 5 gives the main elements of that calculus for a C2-submanifold of R N of codimension 1 including Stokes's and Green's formulae in section 5.5 and the relationship between tangential and covariant derivatives in section 5.6. This is applied to the derivative of the integral of the square of the normal derivative of section 4.3.3. Section 6 extends definitions and structure theorems to second-order derivatives. In order to develop a better feeling for the abstract definitions the secondorder derivative of the domain integral is computed in section 6.1 using the combined strengths of the shape and tangential calculi. A basic formula for the second-order 329
330 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus semiderivative of the domain integral is given in section 6.2. This is completed with structure theorems in the rionautonomous case in section 6.3 and in the autonomous case in section 6.4. The shape Hessian is decomposed into a symmetrical term plus the gradient acting on the first half of the Lie bracket in section 6.5. This symmetrical part is itself decomposed into a symmetrical part that only depends on the normal component of the velocity field and a symmetrical term made up of the gradient acting on a generic group of terms that occurs in all examples considered in section 6.
2
Review of Differentiation in Banach Spaces
In this section we review some elements of derivatives and semiderivatives in vector spaces. In that context we shall start with the weaker notion of Gateaux semiderivative and emphasize the key role played by the Hadamard semiderivative in the semiderivative of the composition of two functions, since in most applications the extension of the chain rule is a central ingredient of a good differential calculus.
2.1
Definitions of Semiderivatives and Derivatives
Definition 2.1. Let / be a real-valued function defined in a neighborhood of a point re of a topological vector space E. (i) We say that / has a Gateaux semiderivative at a point x 6 U in the direction v G E if the following limit exists:
when it exists it will be denoted by df(x; v). (ii) We say that / has a Hadamard semiderivative at x e U in the direction v G E if the following limit exists:
when it exists it will be denoted by dfjf(x;v). The above definitions extend from a real-valued function to a function / into another topological vector space F. It is clear that df(x; v} exists and is equal to d n f ( x ; v) whenever the semiderivative dnf(x;v) exists, but the converse is not true without additional assumptions, as can be seen from the following example.
2. Review of Differentiation in Banach Spaces
331
Example 2.1. Consider the function / : R2 —» R:
It is readily seen that / has a Gateaux semiderivative at (0,0) in all directions v in R and that
which is trivially linear and continuous with respect to v. However, if for e > 0 we choose the directions
then
and djjf(Q, 0; 1,0) does not exist. We shall also need the following notions of full derivatives. Definition 2.2. Let / be a real-valued function denned in a neighborhood of a point x of a normed vector space E. (i) / has a Gateaux derivative at x if Vv 6 E, df(x;v) exists and v i—> df(x; v) : E —> R is linear and continuous.
(2-5)
Whenever it exists the linear map (2.5) will be denoted V/ : E —» E'. (ii) / has a Frechet derivative at x if it has a Gateaux derivative at x and
where (-,-}E denotes the duality pairing between E' and E. The above definitions extend from a real-valued function to a function / into another normed vector space F. The Gateaux semiderivative is generally neither linear nor continuous with respect to the direction, even in finite dimension.
332 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus
Example 2.2.
Consider the following function / : R2 —> R:
It is Gateaux semidifferentiable at (0,0) in all directions v = (vi,v%) in R2, and
is neither linear nor continuous with respect to v. As for the Gateaux semiderivative the Hadamard semiderivative is generally neither linear nor continuous with respect to the direction.
Example 2.3.
The Hadamard semiderivative of the norm f(x} = \X\E at x = 0 is given by
It is continuous but not linear in v. When / has a Gateaux semiderivative in all directions v at a point x that is continuous with respect to v, it has a Hadamard semiderivative in all directions
but again it is not necessarily linear in v. 2.2
Locally Lipschitz Functions
The following theorem gives a sufficient condition for the equivalence of Gateaux and Hadamard semiderivatives (resp., Gateaux and Frechet derivatives). Theorem 2.1. Let E be a normed vector space. Given a function f : U —> R which is uniformly Lipschitzian in a neighborhood U of x in E, that is,
(i) If df(x;v) exists, then dnf(x',v} exists and df(x;v) = dfjf(x;v}. Moreover, if df(x; v) exists for all v £ E, then
(ii) If f has a Gateaux derivative at x, then it has a Frechet derivative at x and they are equal.
2. Review of Differentiation in Banach Spaces
333
Proof, (i) There exist s > 0 and a neighborhood W of v in E such that
Then
and
So as e —» 0 and w —> v, d#/(ar, t>) = d/(x; v). (ii) This part of the proof is obtained from Lemma 3.1 in Chapter 4 extended to a normed vector space. D
2.3
Chain Rule for Semiderivatives
The main difference between the two semiderivatives is that the composition of two functions that are semidifferentiable in the sense of Hadamard is semidifferentiable in the sense of Hadamard. Theorem 2.2. Let E and F be two topological vector spaces and let h be the composition of two mappings f and g:
in a neighborhood U of a point x in E, where
Assume that (i) g has a Gateaux (resp., Hadamard) semiderivative at x in the direction v, and (ii) d H f ( g ( x } \ d g ( x ] v } }
exists.
Then
Proof, (a) For 5 > 0 small enough let
334 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus
By assumption, m(e) —> 0 as e —> 0. By the definition of dh(x; v) we want to find the limit of the differential quotient
which can be rewritten as
where So by the definition of du /,
(b) When g is Hadamard semidifferentiable we replace m(e) and d(e) by
and
and proceed as in part (a). In general we cannot improve the semiderivative of h by improving the semiderivative of g when / is not Hadamard semidifferentiable.
Example 2.4.
Consider the composition of the function / : R —> R in Example 2.1 and the map
The map / is Gateaux, but not Hadamard semidifferentiable, and the map g is infinitely differentiable. However, the composition
is not even Gateaux semidifferentiable at 0. We reiterate that the Hadamard semidifferentiability is a key property for the "chain rule." In general, the composition h — f o g of two maps will fail to have a semiderivative unless / has a Hadamard semiderivative—even if the map g : E —> F is Frechet differentiable at the point x.
2. Review of Differentiation in Banach Spaces
2.4
335
Semiderivatives of Convex Functions
Finally, the class of functions that are Hadamard semidifferentiable is not restrictive since it contains the classical continuously differeritiable functions and the convex continuous functions. Theorem 2.3. Let f : U C E —> R be a convex function defined in a convex neighborhood U of a point x of a topological vector space E. (i) There exists a neighborhood V of x such that
(ii) /// is continuous at x, then there exists a neighborhood W of x such that
Proof. Part (ii) is a consequence of part (i) and the fact that a continuous convex function at x is locally Lipschitzian in a neighborhood of x (cf. Ekeland and Temam [1, pp. 11-12]) by direct application of Theorem 2.1. We give the proof of part (i) for completeness. Let 0 e ]0,1]. Notice that for fixed x and v,
For this fixed a, we show that
This follows from the identity
and the convexity of /,
This can be rewritten
and yields (2.17). Define
336 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus
Now we show that ^ is a monotone increasing function of 0 > 0. For all Oi and 6-2, 0 < #i < 6>2 < 00:
which implies that ) Consider the quotient
as w —•> v and t \ 0. Hence h(t, w) = tw —•> 0 as w —» v and t \ 0. Then
where
and 0 if h — 0. By assumption, Q(h(t, w)) —>• 0 since h(t, w) = tw —> 0, v = 0, and (Vf(x},w)E —* (V/(x),f) J 5; as w —>• v by continuity. Therefore,
and dnf(x\v] exists and dfjf(x;v} = (Vf(x),v)E is linear and continuous with respect to v. (•£=) Define the element V/(x) of E' as
2. Review of Differentiation in Banach Spaces
337
Denote by Q the limsup of Q(h] as \h\ —» 0, which can possibly be infinite. Notice that for h ^ 0
Since E is finite dimensional, there exists a sequence {hn} and v E E such that
By letting tn = hn\,
Therefore, Q = 0 and / is Frechet differentiable at x. We complete this section with a classical sufficient condition for the Frechet differentiability. Theorem 2.5. Given a normed vector space E and a map f : E —>• R, assume that there exists a neighborhood V(x] of x E E such that (i) for all y E V(x], f has a Gateaux derivative V/(y), and (ii) the map V/ : V(x) —> E' is continuous at x. Then f has a Frechet derivative at x. Proof. There exists p > 0 such that the open ball B(x, p) is contained in V(x). For h £ E, Q < \h < p, consider the quotient
There exists a, 0 < a < 1, such that
This shows that / is Frechet differentiate at x.
338 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus
2.6
Hadamard Semiderivative and Velocity Method
In section 2 of Chapter 7 we have drawn an analogy between Gateaux and Hadamard semiderivatives on one hand and shape semiderivatives obtained by a perturbation of the identity and the velocity method on the other hand. The next theorem relates the Hadamard semiderivative and the semiderivative obtained by the velocity method for real functions denned on R N . Theorem 2.6. Let f : NX -^ R be a real function defined in a neighborhood NX of a point X in R N . Then f is Hadamard semidifferentiable at (X, v) if and only if there exists r > 0 such that for all velocity fields V : [0, T] —> R satisfying the assumptions (a) (b)
(c) the limit
exists and for all V satisfying (a) and (b) and V(0) = v,
where Tt(V}(X]
= x(i) is the solution of the differential equation
Proof. (=£>) Let V be a vector field satisfying conditions (a), (b), and (c). Define
It is continuous on ]0, r] and
Therefore
2. Review of Differentiation in Banach Spaces
339
So
since / is Hadamard semidifferentiable at (X, v). The limit only depends on V(0) = v. ( v the limit
exists and only depends on v. We now show that we can associate with such a sequence {wn} a velocity field V satisfying (a) to (c) such that V(tn,X} = wn and Ttn(V)(X) = X + tnwn. Then from properties (2.18) and (2.19) we conclude that d n f ( x ] v ] exists. Set to = 1, tn = 2~ n , xn = X + tnwn and observe that tn - tn+i = -2~( n+1 ). Define for m > 1 the following CTO-interpolation in (0,1]: for t in [t n + i,t n ]
where j>, QQ, q\ G P 2m+1 [0,1] are polynomials of order 2m + 1 on [0,1] such that p(0) = 0 and p(l] = 1 and p^(0) = 0 = pW(l), 1 < I < m, q0(Q) = 0 = q Q ( l ) , q'0(Q) = 1, ^(1) = 0 and ^>(0) = 0 = 9^(1), 2 < t < m, 9l(0) = 0 = 9l (l), ^(0) = 0, q { ( l ) = I and q[l}(0) = 0 = q [ e ) ( l ) , 2 < i < m. By definition T ( - , X ) G C m ((0,1]; R N ). Moreover, for 1 < ^ < m,
We now show that T satisfies conditions (a) and (b) which are equivalent to condition (4.2) in Chapter 7 on V. First observe that Tt(X] - X and dT/dt are both independent of X and (Tt - I)(X) 6 C m ((0,1]; R N ). Define
340 Chapter 8. Shape Derivatives and Calculus, and Tangential Differential Calculus
Hence T and dT/dt clearly satisfy condition (Tl) in (4.8) in Chapter 7. It satisfies condition (T2) since Tt~l(Y) = Y-f(t), and a fortiori T~l satisfies condition (T3). Therefore, the velocity field
satisfies condition (V) given by (4.2) of Chapter 7. Thus properties (a) and (b) are satisfied. It remains to show that df/dt(t) —> v as t —*• 0. It is easy to check that by construction1 —p'(r) +
