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0. By I, u = u(x0) in a neighborhood of x9. Thus the set of points at which u = u(xo) is open. Since it is closed by continuity, it must coincide with the whole domain 9.
For the proof of IV, assume that k is a point on I with = max u = u(.x) > 0. Then Mu = Lu - au > 0 near z and, by III, either du/d1V > 0 at X or u - µ at some interior points close to X. In the latter case u is constant by II. 2.3.
Applications to the Dirichlet Problem
We shall now discuss some applications of the maximum principle to boundary value problems. Consider first the Dirichlet problem u = O on (3) Lu =f in Vt, where f, 0 are given bounded functions.
THE MAXIMUM PRINCIPLE THEOREM.
153
If a < 0 then the solution u of the Dirichlet problem (3)
satisfies the inequality
Jul < max 101 + (e" - 1) max If I
(4)
where a = (11m) [K -{- (K2 + 4m)112] and d is the diameter of 9.
The interesting thing about this inequality is that it does not depend upon the size of a, or on the continuity properties of the coefficients, or on the shape of the domain.
An immediate corollary of this estimate is the uniqueness theorem for the Dirichlet problem. If f = 0 = 0, a < 0 and (3) holds, then u = 0. Of course, this also follows directly from II. In order to prove the estimate we may assume, without loss of
-
generality, that 9 lies in the strip 0 < x1 < d. Set g(x) = e' eu'. In the strip considered, we have that e"d - 1 > g > 0 and Lg = (a11a2- a1x)e'zl + ag < -mat ; Kx = -1. Now set
-
h =maxI0I +g(x)maxIfI.Then Lh =maxIf(Lg+amaxI0l -max If I and h > max 101. Hence v = u - h < 0 on 9 and Lv=f - Lh >f -}- maxIfI >0 in W. By II, v :!5: 0 in 1.
Similarly, u ! -h. Hence Jul < max 101 - max If I g(x) which implies the assertion. The proof shows that the theorem can be considerably strengthened.
If a is not nonpositive, we cannot expect in general that the Dirichlet problem will have a unique solution. Nevertheless, the
uniqueness theorem can be proved and an estimate can be obtained if the domain is sufficiently small. THEOREM. Assume that a < k where k is a positive number, and that the diameter d of 0. By II, z
must be a boundary point. Near this point Mu = -au > 0 so that the normal derivative of u must be positive at f by III. But
all tangential derivatives vanish at X. This implies that the prescribed derivative au/aa = 0 at x, which is impossible.
THE MAXIMUM PRINCIPLE
155
2.5. Solution of the Dirichlet Problem by Finite Differences
As the next application of the maximum principle, we shall discuss the effective numerical computation of a solution of the Dirichlet problem (3) by the method of finite differences. For the sake of simplicity, we restrict ourselves to two dimensions and to a differential equation of the form
Lu = Du + alu, + a2u,, + au =f
(6)
The functions al, a2 and a are assumed to satisfy the conditions
aS0 We approximate the differential operator L by the finite $a11 + 1a21 < K,
difference operator
Lhu=Mhu+au
(7)
= h-2[u(x + h, y) + u(x - h,y) -{- u(x,y + h) + u(x, y - h) - 4u(x,y)] u(x + h,y) u(x h,y)
-
+ a1(xy)
2h
u x, y
u x, y + a2 (xy)
-
+ a (xy) u (x,y)
2h
the mesh width h being a small positive number. Note that Lhu
Lu as h
0 for every C2 function u.
In order to formulate a boundary value problem for this difference equation, we introduce the following notations. The points
(x + h, y), (x, y + h), (x - h, y), (x, y - h) will be called the h-neighbors of PO = (x, y) and will be denoted by P01, P021 P03, P. Py situated in 9, A lattice domain'h is a set of points Pi, P2,
....
having coordinates which are integral multiples of h, and such that for every point in the set its h-neighbors belong to K We also require that given any two points Qo and Q00 in 'h there should exist points Qo = Q1, Q2, ... , Q, = Q00 in Wh with Qj
an h-neighbor of Q,_1. Neighbors of points of Fh which are not
themselves points of Th form the boundaryh of !Yh; these points will be denoted by PV+1, PV+21 ... , P' ,-sr. The union of
L. BERS AND M. SCHECHTER
156
yh
and 'h is the "closure" 1h of !Ye,,. We assume that h is so
small that a lattice domain 65h exists and that
hK < 0 < 1
(8)
For the difference equation we now pose the following Dirichlet problem (9)
Lhu
=fatP1,P2,...,P.V;
u =0 atP.,+1,.
.
, PV+.11
The solution u is of course to be defined only at the N -i- M points P,.... , Pv h,,, or as we say, on the lattice 7h.
We claim that the following maximum principle holds for functions u defined on the lattice domain h. If Mu > 0 in 1,
then either u is a constant or it achieves its maximum on the boundary The proof is trivial. For a point Pi in F,, the condition 0 reads Mhu
u(Pi)
0 in,, cannot have an interior positive maximum unless it is a constant. Furthermore, the same reasoning which led to the estimate (4) leads now to the following estimate for the solution of the Dirichlet problem (9) : (10)
Jul < max 101 ; C max If I
THE MAXIMUM PRINCIPLE
157
where C depends only on K, the domain T, and 0. In fact, we may use the same comparison function as before since we have that Lh(ead - exz) _ -a2 e,,x(sinh (ah/2))2 ah/2
-blot e"x sinh (ah) + a(e"d - e"s) /2 )12l(1
(sin
- ot2 a
ah/2
J
- K ah coth (ah/2) J < - 1
if a > 0 is large enough, provided that 9 lies in the strip 0
2)
_2
1
J' Au (y) log
x
dy
(n = 2)
yI
where uw,, is the area of the unit sphere in n-space. Computing the gradient of v by differentiating under the integral sign we obtain easily the estimate sup, Igrad vi < cKR,u where c depends only on n. The function v(x) is of class C1 everywhere, and by Green's third identity, fi(x) = u(x) - v(x) is harmonic in A. Now, the derivatives of 0 are harmonic and obey the maximum principle. Hence
sup, (grad I < maxr (grad 01 < v + cKR4u so that
it < sup, Igrad 01 + cKRµ < v -!- 2cKRy If R < Ro < 1 /2cK, we see that 1u < kv, with k = 1/(l - 2cKRo). The convexity of A was used only in asserting that max., Jul < R max, (grad ul ; it would suffice to assume 0 star-shaped with
respect to the point where u vanishes. For nonconvex ., with sufficiently smooth boundary the theorem remains valid, with k depending on A. No convexity is needed if a = 0. In this case the condition that u vanish somewhere in A is also superfluous: it can be achieved by adding a constant to u. The same method yields a maximum principle for an elliptic system in the plane Ox - tVv = x111 - a12V (13)
= x210
x22tV
L. BERS AND M. SCHECHTER
162
with bounded a;;. This is not surprising since for the special case a21 = a22 = 0, system (13) shows that 4 = u,,, tp = -uv where u is a solution of E u + a11um + a12u = 0 Setting 0 + i v = w we may write (13) in the complex form
(14)
w,, =aw±bw
(15)
where a and b are bounded complex-valued functions. This equation implies, of course, the inequality Iw2I < K IwI
(16)
and for such an inequality we have the following THEOREM.
Let w(z) be a function of class C1 in a domain A of diameter
R < R0, satisfying inequality (16), and assume that w is continuous on the boundary 1' of A. Then (17)
supA IwI < k maxi, IwI
Ro > 0 and k > 0 being constants depending only on K.
The proof consists of applying the maximum modulus theorem to the holomorphic function w(z) - r(z), z e A, where (18)
r(z)
iff
w,( z) dz dry
+ i?l
0
and noticing that sup_A Irl < 2R sup,, I wtl < 2RK sup,, IwI. 2.8.
Carleman's Unique Continuation Theorem
The maximum principle just stated implies a celebrated theorem of Carleman. THEOREM. A function w(z) of class C1 which satisfies inequality (16) in a domain V vanishes identically if it has in 9 a zero of infinite order.
Assume that w(z) = 0 (Izl`), z --) 0, for all N > 0. It suffices to show that w(z) = 0 for Izl < R, R > 0 and small Proof.
THE MAXIMUM PRINCIPLE
163
enough. The functions w(z) n = 1 , 2, ... , vanish at z = 0 and satisfy (16). Applying (17) to the domain 0 < I zI < R, we see that max I z-"w(z) I< kR-" max I w(z) I Izj 0, Ht is the space of periodic functions having
generalized L2 derivatives up to the order t.
In particular, a periodic function of class Ct belongs to H. A partial converse of this statement is also true. LEMMA 3.
(Sobolev [1]). If u(x) E H, and t > [n/2] + k + 1,
then u is of class Ck, and (17)
max IDDul < const. Ilullt
for IpI < k
L. BERS AND M. SCHECHTER
168
It will suffice to prove this for k = 0 (cf. (10)). Now if u = E afe'f r c Ht, t > [n/2] + 1, then
Ixfl)I0
these inequalities, combined with (23), yield
(u, Lou + Liu) > ci
11'
t
- cl
(ci > 0)
1 1u1 1u
Case (c). L = Lo + Li + L2 where Lo and Ll are as in Case (b)
and L2 =
aD(x)DP. By integration by parts and by the Ipl <m
argument used before (u,L2u)o = 1J a,qD3uD°u dx
IPI + IQI < m,
I pI < m/2,
IQI < m/2
L. BERS AND M. SCHECHTER
172
so that (25)
(u,L.2u)01
I
const. IIu ll,r,/2 Ilu II (rn/2)-1
>0 For small a this together with (24) yields Garding's inequality. General Case. For a sufficiently small n > 0 we construct e const. Ilu
(e-"'/2) const. Ilu 112,
periodic C. functions aq(x), (u2(x), . . properties. (i) On the support of
.
,
(o).\-(x) having the following
each w, the oscillation of IpI = m, of L is less than 71.
each leading coefficient (ii) E (i)1(X)2 - 1. By Case (c) (26)
((u,u, Lo);u) o > pos. const. II oo;u Il ,,,2 - const.
II (,),u II 2
but (u,Lu) 0
(27)
f
= (I (1 )uLu dx =
((o,u, L(o;u)o ± R
R = -1 f(f))u {L(oju - (u,Lu) dx Integration by parts yields
R=J
dx,
IpI + 191 < m,
with some fixed C,, functions
141 < m/2
and as before (e-m/2)
IRI < e const. Ilu IIn,/l +
(28)
I pI < m/2,
const. Ilu Il0
Clearly II(,),u Ilo pos. const.
II U
II71t/2 - const. Ilu 112
Combining relations (26-30) we get Garding's inequality. The use of Fourier series (or Fourier transforms) seems indispensable for the proof of Theorem 1. But in the classical case m = 2 integration by parts suffices. Indeed, assume that
L-L1 L1
= -5 alk(x)
a2
a.r,ax; '
,
L2 a
L2 = Y a,(x) ax, - a(x)
HILBERT SPACE APPROACH, I
with E aik(x)
173
c E 6rj, c > 0. Then
au au
= SJ a,k ax. axk dx +
(u)Liu)o
a
2
>cI
If ax
au ku
t
dx
x
-con st. Ilullo Ilulll
11TIUxillo
pos. const.
1 1 U1 1i
- const.
11U 11
o
and since I (u,L2u) I is easily seen not to exceed e const. IIu II
(1/e) const. IIu Ilo for every e > 0, Garding's inequality follows.
-
At the moment we shall prove Theorem 2 only in a weakened form. More precisely, we shall show that (22) holds for A > A0(t), A0(t) depending on t and L.
Let s be a fixed non-negative integer. The differential operators K"L and LK" of order m + 2s are elliptic; hence, by Theorem 1, there are positive constants c;, c., depending on s such that Proof.
(u,K"Lu)o
:2! c'l Ilu ll8+,,,/2
(u)LKsu)o ? c1
IIu
112
- C2 Ilullo - c2' IIu Ilo
Using the first inequality together with (15, 6, 14, 4, 3) we have that flu II 8 I m/2
II Lu + Au II 8-m/2 =
IIuII "+m/2
II K3Lu + AK"u ll -s-m/2
> (u, K"Lu + ).K"u)o >- ci IIUII8+m/2 - c2 Ilullo + A(u,Ksu)o
=
ci flu Iii+m/2
- cz IIu Iln + )'11U112 ? ci
IIu IIS+m/2
+ (2 - c2) Ilullo ? ci Il u ll9 i m/2 for A > c2 and dividing by 11U 113. m/2 we obtain (22) for t = s + m/2. Similarly, IIuII-s-m/2
IILu + Aull_s-m/2 = IIK-'ulls-,,,/2 IILu + Aull-s-m/2
> (K-"u, Lu + Au) o = (K-"u, LK"K-"u + Au) IIK_"u112
ci
= ci
11K-"ull8+m/2 - c2
IIUI12
8+m/2 - c2
IIUII
+ A IIuI12-8
-2" + . IIuII? ? ci Ilullo 8+m/2
for A > c? and dividing by IIu II -s 1 m/2 we obtain (22) for t = -s + m/2.
A complete proof of Theorem 2 will be given later.
L. BERS AND M. SCHECHTER
174
3.5.
Differentiability Theorem
Theorem 2 (in its weakened form) implies LEMMA 11.
For every t and for A > 0 sufficiently large the operator
L + A is a bounded linear one-to-one mapping of H, onto the whole of H,-,n and the inverse mapping (L + 2) -1 of H,_m onto Ht is also bounded, independently of A.
Proof.
By Lemma 9 the linear mapping L + A: Ht -* H,-,. is
bounded. If 2 is so large that (22) holds (at the moment this choice may depend on t), then Lu + Au = 0 implies u = 0, so that the mapping is one-to-one. The inverse mapping is defined on
the range R, = (L + A) (H,) and is bounded by virtue of (22), the bound being independent of A for A > A0(t). Next, R, is closed. Indeed, assume that vi E R, and II v, - v 0. Then v! = Lu, + Au,, u, e H. By (22) we have that II u, - uk II t S c3(t) II Vi - Vk II t-m , 0 for j, k ---+ oo. Hence there is a u-E Ht such
that Ilu, - u II, -> 0. By Lemma 9, IILu1 + Au, - (Lu + Au) II t-m
0. Hence Lu + Au = v and v e R. In order to prove that R, = H,_,n we use the adjoint operator L* (cf. Chapter 1). We note that (31)
(u,Lv)o
- (L*u,v)o = 0
for periodic C,, functions (proof by integration by parts) and hence
(by an obvious limiting argument) also for u c- Hs, v e Ht with
s+t>m.
Assume that A is so large that (22) holds also for L* and that R, 0 H,_r. Then there is a w e H,_,n such that w 0 0, (w, Lit + Au),-,. = 0 for all u in Ht (projection theorem, cf. Appendix I to this chapter). Hence 0 = (Kt-mw, Lu + Au)o = (L*Kt-mw + AKt-mw, u)o for all u in H,, and in particular for all C,,, functions u. This implies
(for instance by (31)) that L*Kt-mw + AKt-mw = 0. Now, Kt-mw e Hm_, and if A is large enough, (22) applied to L* shows
that K'-mw = 0. Then w = 0, contrary to our assumption. THEOREM 3. If u is a periodic distribution and Lu a H u EHs+m
then
HILBERT SPACE APPROACH, I
175
This is a differentiability theorem. It implies that every (periodic) solution of Lu =f is a function if f E H_,,,, has L2 derivatives up to the order k if f e Hk_,,,, is of class C, if f e Hs with s >_ [n/2] + r + 1 - m (Lemma 3), and is of class C,,,, iff is. In the next chapter we shall extend this result to nonperiodic solutions.
Set Lu ==f. By Lemma 7 the distribution u belongs to Hk for some k. Hence f + Au E Hm1n (k,8) and if A is large enough u = (L + A)-' (f + Au) belongs to Hmin (k+m,s+m) Proof of Theorem 3.
Repeating the argument we see that u c- Hmin (k+im,s+m) for j = 1, 2,.... Thus u c- H,,+).,,. 3.6.
Solution of the Equation Lu = f
We shall show now that the equations (32)
Lu =f
(33)
L*v =f
form a Fredholm pair.° Here f is a given periodic distribution;
we already know that for f EH, every (periodic) solution belongs to He. In particular, all solutions of the homogeneous equations
Lu = 0 L*v = 0
(34)
(35)
are Cr,, functions.
Let A0 > 0 be so large that the bounded linear mappings M = (L + AO) -1: Ho H,,, and M* = (L* + An)-1: Ho --* H. exist (Lemma 11). Since H,,, c Ho we may, and shall, consider
M and M* as mappings of Ho into itself. By Lemma 10 these mappings are completely continuous. The homogeneous equations (34, 35) may be written in the form
u-k0Mu=0, 6 See Section 1.5.
u-A0M*u=0
L. BERS AND M. SCHECHTER
176
Furthermore, the operators M and M* are conjugate in Ho. Indeed, by (31), we have that (Mu,v)o = (Mu, L*M*v + AOM*v)o
= (LMu + ).0Mu, M*v)o = (u,M*v)o
Hence the Fredholm-Riesz-Schauder theory is applicable (cf. Appendix II to this chapter). We obtain THEOREM 4.
The homogeneous equations (34, 35) have the same
finite number of linearly independent solutions. THEOREM 5.
The equation Lu + Au = 0 has nontrivial solutions
only for a denumerable set of values ) with no finite accumulation point.
Indeed, the equation considered may be written in the form
u-A'Mu=0,with 2'=io-2. THEOREM 6.
Equation (32) is solvable if and only if (f,v) o = 0
for every solution v of (35).
Assume first that f e H, for some s -,,>- 0. Then equation (32) is equivalent to the following equation in Ho: Proof.
u - AOMu = Mf By the general theory, it is solvable if and only if (Mf,v)o = 0 whenever v - AoM*v = 0. But this is the same as saying that (f,v)o = A0(f,M*v)o = 2o(Mf,v)o = 0 whenever v - ,10M*v = 0. Since v - A0M*v = 0 is equivalent to L*v = 0, the result follows in this case.
Assume now that f e H , for s > 0. Let t be an integer such
that m + 21 > s. Since LK' and K'L* are adjoint elliptic operators of order m + 2t, there is a Al such that LK' + ,11 and K'L + Al are one-to-one linear mappings of Hm±l,_, onto H ,. It is easily Set M' _ (LK' + 21) -1, M'* = (K'L* + shown, as in the discussion preceding Theorem 4, that M` and M'* are conjugate in Ho. We now note that (32) is solvable if and only if (36)
LK'w = f
has a solution. Now (36) is equivalent to (37)
w - a1M'w = M !f
HILBERT SPACE APPROACH, I
177
and My E Hm+2t-s c Ho. Hence we may again apply the abstract theory to conclude that (37) has a solution if (M f v)o = 0 for all v E Ho such that v - AIM(*v = 0. This is the same as saying that (32) has a solution if (f,v)o = A1(f,Mt*v)o = A1(M f,v) = 0 for all solutions v of (35). Thus the theorem is proved in this case as well.
Combining Theorems 4 and 6 (or reasoning directly from the general theory) we get Equation (32) is solvable for every f c- H ,, if and only if (34) has only the trivial solution u = 0. THEOREM 7.
We can now complete the proof of Theorem 2. We first note that from Garding's inequality (21) it follows that for some A > 0, Lu + Au = 0 implies u = 0 whenever A > A [e.g., take A = c2 in (21) ].' Now assume that Theorem 2 is false. Then there exists, for some fixed t, a sequence {u5} in Ht and a sequence of numbers A; > A such that II u; IIt = 1, IILu, + A,uf llt_m , 0. Let Ao be so
large that the bounded mapping (L + A)-1: Ht_.,,, , Ht exists for 2 >_ Ao (Lemma 11). Clearly A, < A. and we may assume, A >_ A. By Lemma 10 selecting a subsequence if need be, that A, we may also assume that {u,} is a Cauchy sequence in Ht_,,,. Now
ui = (L + Ao)-1(Lu, -- A,u,) + (A0 - 2,)(L + Ao)-lu1 so that 0 for j, k -- oo. Hence there is a u in Ht with II us - Uk Il t 0 and we have : Il u ll = 1, Lu + Au = 0 which is 11u1 - u II t impossible.
The same method shows that whenever A is not an eigenvalue
(i.e., whenever Lu + Au = 0 implies that u = 0) (L + A) -': Ht is a bounded operator, the bound being uniform Ht_m on every closed set of A's not containing eigenvalues. Appendix I. The Projection Theorem
We shall consider some simple theorems in a real Hilbert space H. (H consists of elements u,v, ... , for which the operations of addition and multiplication by real numbers are defined and these 7 Recall that Lu + ..u = 0 implies, by Theorem 3, that u is a C,, function. Hence the GArding inequality can be applied.
L. BERS AND M. SCHECHTER
178
operations obey the usual rules of a vector space. Moreover, for every pair of elements u,v there is defined a real number (u,v) called the scalar product in such a way that (u,v) = (v,u), (u + v, w) =
(u,w) + (v,w), (Au,v) = A(u,v), (u,u) > 0 for u ; 0 for all u, v, w c- H and real number A. In such a case we can define a norm 11u II = 1/(u,u), and it is easily shown that (u)v)I S IIuII IIuII 11u + v11 0 as m, n -- co, there is a u E H such that 11 u,, -u11 -*0asn- oo.) A subset S of H is called a subspace if Au + ,uv is in S for all real numbers A, a whenever u and v are in S. It is closed if ueHand Ifu, -u11 ->0asn- imply that u E S. First we prove
c S,
Let M be a closed linear subspace of H. Then for every u c- H not in M there is a v e M such that LEMMA 1.
IIu - v 11 = g.1.b. 11u - w II WEM
Let d = g.l.b. flu - w 1j, w c- M. Then there is a oo. From (1) sequence c M such that IIu - w, II , d as n Proof.
we see that 4 Ilu - (wm + wn) III + IIWm - wnII2
=
2( IIu
- w. 112 + Ilu -wn 112) -' 4d2
as m,n -> oo. Since-(w,. + w,,) c- W, 4 IIu
-
(wm + wn) II2 > 4d2
- wn II -' 0 as m, n -> or,. Since H is complete, there is a v c- M such that 11 wf, - v 11 - 0. This means that and hence
11 wm
IIu - v 11 = lim flu - wn 11 = d.
HILBERT SPACE APPROACH, I
179
(Projection Theorem). Let M be a closed linear subspace of H. Then for every u e H, u = v + w, where v e M and (w,M) = 0 [i.e., (w,h) = O for all h e M]. THEOREM 1
Proof.
If u c- M, set v = u and w = 0. If u is not in Al, then
by Lemma 1 there is a v e Msuch that Ilu - v II = d, the "distance" from u to M. Now if f is any element of M
IIu-vll2-22(u-v,f)+22Ilfii2
IIu - L'IIZC llu - U - ?/ II2= for all real 2, in particular, for
2=(u-v,f) Ilf112
Thus
Ilu - vl12 < Ilu - vll2 -
2(u
IIf112
f)2 +
(u
f
II,f)2
and hence (u - v, f )2 < 0, which means that (u - v, f) = 0. Since f was any element of M, w = u - v meets the requirements of the conclusion of the theorem. COROLLARY 1.
If M 0 H, there is a nonzero element w of H such
that (w,M) = 0. Proof.
Take u not in M. Then by Theorem 1, u = v + w,
where v e M and (w,M) = 0. Clearly w
0.
A bounded linear functional Fu on H is a real valued function on H
which satisfies the following conditions: F(ui + u2) = Ful + Fu2; F(2ul) = 2Fui, 2 a real number; IF(u) I < K IIu II for all u e H and some K > 0. The smallest K for which the last statement holds is denoted by IIFII
THEOREM 2 (Representation Theorem; Frechet, Riesz). For every bounded linear functional Fu on H there is a uniquely determined element f e H such that Fu = (u, f) for all u e H and IIFII = IIf1I Proof.
Let N be the set of all v e H such that Fv = 0. N is a
linear subspace of H. For if vl and v2 are in N and Al and 22 are any real numbers, F(2ivl + 22v2) = ).1Fvi + 22Fy2 = 0. Moreover, Nis closed in H. For if {v,,} is a sequence in N and II v,, - v II -0, theniFvl and V EN.
180
L. BERS AND M. SCHECHTER
Now if N = H, the theorem is easily proved by setting f = 0. Otherwise there is a w 0: 0 in H such that (w,N) = 0 (Corollary 1). Therefore Fw - 0 and for any u E H
FIu
-Fwwl =Fu -FwFw =0
and hence u - Fw w is in N. This means that
Iu -F-tww,wJ
=
0
i.e., that (u, ') =Iw 11w112
Therefore
Fu=
(u
w Fw)
' IIwII2
1
and (w/III' 112)Fw is the element f in the conclusion of the theorem. If f' also represents F, then Il f _f'112 = (f - f ', f -fl) _
(f -f',f) - (f - f',.f ') = F(f -f') - F(f -f') = 0. Next, Ilf III = (,f f) = F(f) IIFII Hence IIFII - E < (g,f) < Il f ll. Thus IIFII = 11f 11.
- E.
The following theorems will be found useful later. THEOREM 3 (Lax-Milgram [1]). Let [u,v]
real valued function defined for pairs of elements in H which is linear in both u and v. Suppose it satisfies (2)
I[u,v]I < lull
(3)
Ilu 112
be a
IIvII
< K[u,u]
for all u, v in H. Then for every bounded linear functional Fu on H there is an f c H such that
Fu = [u,f] For fixed v e H, we have, by (2), 1 [u,v] I < const. II u II Hence [u,v] is a bounded linear functional in H. Thus (by Theorem 2) there is an element Sv in H such that Proof.
(4)
[u,v] = (u,Sv)
HILBERT SPACE APPROACH, I
181
Obviously, Sv is a linear mapping of H into itself. We observe that by (2) IISuII 2 = (Su,Su) = [Su,u] S IISuII
.
Hull
so that 11sull
0 such that lITull 0 such that
cl' Ilull Tu and hence u = Tu. But this means that u e N. Since u is also in M, u = 0. But lull =
= 1, giving a contradiction. For any fixed v e H, Lemma 4 shows that
lira llu1II
11v 1111 Tu 11
m
189
must have
Amvm) = v - (v - A,,Tv,, + AmTvm) and
Amvm)II2 = IIvn112 + IIv - A,,Tv + AmTv,,, II2 > 1
since both v - A Tv and Tum = L- are in Ei_1. This shows that m
has no convergent subsequence. Hence there does not exist a bounded sequence of distinct 1.,, and the theorem is proved.
CHAPTER 4
Hilbert Space Approach. Dirichlet Problem 4.1.
Introduction
We saw in the preceding chapter that the study of periodic solutions of elliptic equations lends itself to a particularly elegant treatment. Some of this machinery can be converted to give weak (or even "semistrong") solutions of boundary value problems and to establish interior regularity. But when it comes to proving that such solutions are smooth up to the boundary, new methods must
be devised. Several authors have studied this problem in great detail (Nirenberg [1], Browder [2], Aronszajn-Smith (cf. Lions [2] ), Schechter [5], Agmon [2], Peetre [1], and others). In this chapter we return to nonperiodic operators
L = I a, (x) D9 IPI <m
defined in a bounded domain V. We assume that L is elliptic and (for the sake of simplicity) that the coefficients a,(x) are in C., in the closure ci of W. We shall study the interior differentiability of weak solutions, formulate the Dirichlet problem within
the framework of Hilbert space, prove the existence of weak solutions and finally take up the problem of regularity at the boundary. 4.2. Interior Regularity
One property which follows easily from the results for periodic equations is interior regularity. THEOREM 1.
Let T be a distribution solution of L T = f in W. If f
is locally square integrable, then T is a function having L2 derivatives of order m. If f has L2 derivatives up to order t, then T has L2 derivatives up to order m + t. 1f f is infinitely dierentiable, so is T. 190
HILBERT SPACE APPROACH, II
191
First assume that T = u, a locally square integrable function. Then u is a weak solution of Lu =f. Since differentiability is a local property, it is sufficient to prove that in each subdomain 91 such that 91 c 9, u has L2 derivatives up to Proof.
order m + t, whenever f has L2 derivatives up to order t, If 91 is such a domain, let cc2 and 93 be such that cc1 92 C 93, cc3 C9. Let be any test function in g3 with in g2. Since u is a weak solution of Lu = f, we have f9, fo dx
t >_ 0.
q2,
=1
uLi O] dx
dx
for all test functions 0 in 9, where Li is a partial differential operator of order < m with C,,, coefficients having support in cJ3. Since all the integrands have support in c3, (1)
fiifw dx
=12
uLi w] dx
for all C., functions w which are periodic in some cube 2 containing T. We can alter u, f and the coefficients of L outside T. so that they become periodic in 2 in such a way that u and f remain in L2 while L remains elliptic having C., coefficients. and the coefficients of Li may be extended without Next, alteration. Denote the periodic extensions by u, f, L, etc. We have A
by (1)
A
A
A
f'=Lv+L,u where v = tic and L, is the formal adjoint of Li . Now by the theory of the preceding chapter f - L1u is in H,_,,,. Hence, by Theorem 3 of Chapter 3, v e H1. This means that u has L2 derivatives of the first order in cc2 '*,'Interpolating a domain between Y1 and cc2 and repeating the process in we see that , f - L,u is in H2_,,, and hence u has L2 derivatives of the second
order in a domain containing .11. This may be continued until L2 derivatives of order m + t are reached.
Assume next that T is a distribution. We shall show later (Corollary 2) that there is a positive integer k and a square integrable function u such that (2)
T[4]
=5
uAk4
dx
L. BERS AND M. SCHECHTER
192
for all test functions 0 in 4j. Assuming this for the present, we have
Jfc6 dr = LT[t¢] = T[L*¢]-J uYL*0 dt
for all such j. Since the operator LA's _ (AkL*) * is elliptic and of order m - 2k, we have by that part of the theorem just proved that u has L2 derivatives up to order m + 2k ; t. Hence by (2)
T[0] =
J
z
uq dx
for all test functions 0. Hence T = AAu and has L2 derivatives of order up to m ± t. The last statement of the theorem follows from Sobolev's lemma (Lemma 3 below). 4.3.
The Spaces H' and H'
We shall now describe some Hilbert spaces which will take the
places of the spaces H, of Chapter 3. For integers t > 0 and for C. , functions in ( we shall employ the scalar products (U) V) t = I J
dx
API v 11 0, II Set Vv = 0v/v II 0Y II Then for every fixed s, II ip, II s = I10, II 8/v 11 0, II v - 0
as v -> cc. By Lemma 3, this implies that Dip - 0 uniformly in Tr for each p. Hence T [v,v] -- 0. But I T [V,] 1, > 1, a contradiction. COROLLARY 2.
For every distribution T there is a non-negative
integer t and a function u e Ho such that
f
T[¢] = guL1/ dx for all test functions 0 in 9. Proof.
By Lemma 6, there is a t ;.>_ 0 such that (4) holds.
Therefore T is a bounded linear functional on the Hilbert space H. Moreover, by Corollary 1, we may consider I It as the norm in Ho. Thus by the Frechet-Riesz representation theorem (Theorem 2 of Appendix 1 to Chapter 3) there is a function u in Ho such that T[0] = (u,o)t
196
L. BERS AND M. SCHECHTER
for all test functions 0. Integration by parts gives
T[O] =J uL's6 dx the required representation. Note that Corollary 2 was employed in the proof of Theorem I. We end the section by proving a simple consequence of the Banach-Saks theorem (Theorem 4 of Appendix I to Chapter 3). LEMMA 7. If ' v is a function in H°, and there is a sequence vn of functions in H' such that
II U. -V 110 - 0,
!IUnii, 0 and set X" > 0
v1(x',xn) = v(x',xn)
_
8-}-1
x,, < 0
.Z.v(x', k=1
... , xi_,) and s = max (m, t ± 1). The constants
where x' = (x,,
Ak are so chosen that v1 is in C. in 2, i.e., that RTL
(21)
k=1
j = 0, 1, ... , S
(-k)'2,, = 1
Thus v1 has compact support in .2, and ID'vlz dx
ID11v1I2 dx
m 1), it follows from (22) that (20) holds for v.
-
It remains to prove that for t < m - 1
,, j h > 0. Letting a -o 0, we have Set s = max (m, t , 1) and suppose that v is in C, n HO in E and vanishes near all faces except x = 0. Assume that D,v E Ht in for i = 1, ... , n - 1, and that IILvll,,,-t+1 is finite. COROLLARY 3.
Then v E H`+1 in 2.' and (26) holds.
We use induction on I and show first that the assertion is true for t = m'. Let a' denote any difference quotient in the x, direction, i , n. Then Proof of Theorem 9.
L,,,r
L8;'v
where L,,, is the operator whose coefficients are the difference quotients of those of L. Thus IILb;'z'il
,n
IIa (Lz')'1 0 (cf. the end of
Appendix II to Chapter 1). Hence, by Lemma 7, D,v E Hm' in S2,_ and IID,vllm' s C'
Letting E -- 0, we see that D,v E H7" in Q,,. Once this is known, we
can apply Corollary 3. We therefore conclude that v e Hm'+i in QT.
Now assume that the theorem is true for t - 1 > m'. We have
LD,v = D,Lv + L'v where L' is an operator of order <m. Hence IILD,vII:-m r + h. Now by Sobolev's inequality (cf. Lemma 3 of this chapter) there is
an integer N (>m) such that max IvI < x(n,N,R)
(4)
Izi _x
Employing the fact that Ivl g,1; and IIVII:,R are equivalent norms
for test functions (Lemma 5 of this chapter), we see that (3) implies (5)
C Const. (ILvi \.-m,r
IIVII.\',r+h
!h
+
Ivl.V-.,,+h)
Now if Isl = N - m, there are C,, functions bxo such that
DBLv = D3Lw + Hence (6)
.1 p'to
bD,D.Dv'4DD"w
N
II wll v,r-h
'fi(x)
-y1)-2 d>'
J`M;)
fo, IX
-1
n-2
an ix ac(y)
1
an
Ix - yl,l-2
dS
dS
Here I is a domain with a sufficiently smooth boundary x is a point of 1, ajan represents differentiation in the direction of the outward normal, and dS is the area element on k For a function
of compact support, we may take 9 to be a sufficiently large sphere. In this case the boundary terms disappear and identity (1)
results. This identity can be rewritten, in terms of distribution theory, as
(n > 2) Oc2 log IXI = b(x) (n = 2) where b(x) is the symbolic Dirac function, i.e., a distribution defined by the relation Ac.,
Ix12-n
= b(x)
J(x)6(x) dx = 0(0) for every test function O(x). Now (la) follows immediately from (1), since O(x - z) dz = f ao(x - z) o r 0(y) dy = ASJ dz x
IX
IZI1--2
-yln-2
=f AM ') IX
yj 11
2
dy
J
IZIn-2
= c 'fi(x)
From (1) we obtain, integrating by parts, the identity (2)
O(x) = const.
y' a4 dy
f X` _ Ix J -yI" ay,
which represents a function with compact support in terms of its derivatives. For an elliptic operator L of order >2 with constant coefficients
and only highest order terms, one can construct explicitly a
POTENTIAL THEORETICAL APPROACH
213
function J(x) related to it in the same way as (xlz-n is related to the Laplacean A. More precisely, the function J (x) will have the following properties. (i) J (x) is a real analytic function for I x I 0. (ii) If n is odd or if n is even and n > m, then
(3)
J (X)
I(x)
where w(x) is positive homogeneous of degree 0, (w(lx) = w(x), t > 0). If n is even and n S m, then
J (x) = q(x) log
IxI +
I(x)
where q is a homogeneous polynomial of degree m - n. (iii) The function J (x) satisfies the equation
L0J (x) = b (x)
(4)
so that for every C., function with compact support (5)
f
f
0(x) = [Lo/(y)]J(x -y) dy = Lo s6(y)J(x -y) dy
The result is classical; we carry out the construction only for the case of an odd n > m and for m = 2 and all n. (As a matter of fact, we shall assume that n is odd and exceeds m whenever it is
convenient to do so during this chapter. This assumption, howdenote the characteristic form of ever, is not essential.) Let Lo, i.e., let a,,DP, I a,E" Lo = IPI =»6
IP1 =m
Consider the function
Ix = x1E1 + ... +
It is obvious that this function is positive homogeneous of degree m + 1 and real analytic for where x -
x 0 0. It is easy to see that Lol(x) is positive homogeneous of degree one and invariant with respect to any rotation about the origin. Hence LOI(x) = const. Ixt. Now set J(x) = cAn+112I(x), where c is a constant. This function has properties (i) and (ii).
L. BERS AND M. SCHECHTER
214
That it also has property (iii) for an appropriate choice of the constant c follows from the fact that L0J(x)
= Loci"+1/2I(x) = cA"+1I2LOI(x) = c0"-1/2 const. IxI = const. A(An-1I'2 IxI)
= const. A x12-" = b(x)
When m = 2 we can write down the fundamental solution just as easily. In fact if
=.I a,,$,s, let A denote the cofactor of a1, in the determinant la,,l. Then
J(x) = c(,J
A.1x,x,)2-,7/2
J (x) = c log (I A1,x,x,)
n>2 n=2
One readily verifies that these expressions satisfy the required stipulations.
The function J (x) just constructed is called a fundamental solution of the differential equation Loo = 0. More generally, for any homogeneous elliptic equation L¢ = 0, a function J (x,)r) depending on a parametery is called a fundamental solution if it satisfies the equation L.J (x,y) = b(x - y)
It should be remarked that bounds on the derivatives of w (x) in (3) depend only on n and the ellipticity constant of Lo, i.e., on
min IQ( ) IFI=1
Fundamental solutions for elliptic equations have been constructed by many authors, the most important results being due to John whose book [1] also contains an extensive bibliography. In
particular, fundamental solutions are known to exist for any equations with analytic coefficients (in which case the fundamental solution itself is analytic) and for any equation with C,o coefficients (in which case the fundamental solution is itself C.) as well as under much weaker assumptions. As a matter of fact, it
is probable that the fundamental solution can be constructed
POTENTIAL THEORETICAL APPROACH
215
for any equation with Holder continuous coefficients, though as far as we know, this has never been proved in the literature.
Fundamental solutions play an important part in the theory of elliptic equations. In particular, once a sufficiently nice fundamental solution has been constructed, one can easily prove analyticity and differentiability of solutions. An arbitrary solution
of an equation defined in a domain 9 with a sufficiently nice boundary ! can be expressed in terms of its Cauchy data at the boundary (values of the function and its derivatives up to the order m - 1) in terms of the fundamental solution of the adjoint equation. If a fundamental solution satisfies appropriate boundary conditions (in which case it is called Green's function), a solution can be expressed in terms of fewer boundary data. Fundamental solutions also play a part in the integral equation approach to boundary value problems. This method is classical for the potential equation and for second order equations in general;
an excellent account will be found in the book by Miranda [1]. The integral equations method has recently been extended to
equations of higher order (Agmon [4] and others). But our concern here will be only with the use of the simplest fundamental
solutions, namely the functions J (x) constructed above. These
applications are based on a simple device due to Korn and Lichtenstein.
To an elliptic operator of order m with variable coefficients a,(x) and to every point x0 we associate the "tangential operator"
=
v
lpl=yn
which is a homogeneous operator with constant coefficients. We shall denote by J.,°(x) the fundamental solution of the equation Lx°4 = 0 constructed above. JZ° is called a parametrix for the equation Lc = 0 with singularity at x0. We shall denote by Sx0
the operator which takes a function O(x) into the function ti = Sx00, where (7) V (x) =fix°(x -y)o(y) dy Finally, we define the operator Tx° by the equation (8)
Tx° = SX°(LZ° - L)
L. BERS AND M. SCHECHTER
216
Since S 0L, ,
= 1 (= identity) on functions with compact support
and similarly L 0ST0 = 1, by virtue of identity (5), we have the following If 0 has compact support, then
"LEMMA" A.
= Tx0 ± Sx0L¢
(9)
and if 0 = Txa¢ ± Sxu f
(10) then
is a solution of the equation Lc = f.
The quotation marks around the word lemma are in recognition
of the somewhat cavalier way in which the lemma has been stated. It goes without saying that in order to become a real lemma,
it must be supplemented by a statement on the assumptions involved. In applications to be considered, all this will be selfevident. The proof of A is clear. Equation (9) follows by noting that on
functions with compact support T,0 + S= L = 1. On the other hand, assuming that (10) holds we have (Tx0 ; Sx L)o = Tx0o -Sx0 f hence, S_L4 = S.,, ,f and Lq = L.,,, S,.0L0 = L,,Syo f =f. The applications of Lemma A are based on certain inequalities which are best stated in terms of certain norms. Some Function Spaces
5.2.
We consider first C X functions defined on the closure of a boun-
ded domain1i. For these functions we set
0,
(jIvdx)
1/r
(r > 1)
l.u.b. ICI
H..gr[o] = l.u.b. ,._0
Ix, - X "l
0 < x < 1
(diam V)' max IIDp0!Ic,(9), In!
k = 1, 2, .. .
POTENTIAL THEORETICAL APPROACH II
(diam V)k-" max H,,,,[Dp0],
IICktx() = It'IICk(l)
1PI =k
k=o,1,..., II0IIJi ()
Il
217
=
(diam
llNk(W) = I (diam q)Iv i -,nlr IPlsk
0<x 0) and is a C. function for x (22)
w(x) dS = 0
J IxI =1
0, and if
POTENTIAL THEORETICAL APPROACH
223
where dS denotes the area element on the unit sphere. (The smoothness condition on w can be considerably relaxed while the condition (22) of vanishing mean value is essential.) It is easily seen that (22) is equivalent to (22')
K(x) dx = 0
I
R1 R then for 0 < x < 1 11K * ¢ II C2(R)
C 0(1)
Applying the Calderon-Zygmund inequality to (26) we have for IPI =m (29)
Rm-nI4 IID1%11L°(R) C o(1)
For 0 < IpI < m we estimate q > n. We have that
II Dny II C,, (R)
using the assumption
I D"J (x) I C const. lxI m-n-IDI
so that for Ixi < R ID"A
const.
Irm-n-IDI
* Iv'l
I
1 /q'
< const. 11+P11L°(R)
)( 1
Ix
l IvI n) (38)
IIUIIR",(RI)
into C
)
and
II (o - L) u ll c« (9) < const. II u II c:+«(V)
so that N maps C2+x (T) into itself, and by (44) IINII = l.u.b.
IINuIIra+s+«(g)
=f, u(') = 0 on #. By Theorem 5 IIUW IIc:+«(g) < C1
(a constant independent of j )
By Lemma 2 we may assume, selecting if need be a subsequence,
that u(' converges uniformly, together with its derivatives of orders 1 and 2 to a function u. Clearly u belongs to C21 «(g), vanishes on the boundary and satisfies the equation Ltou = f. Hence to belongs to -r. In order to solve the nonhomogeneous boundary value problem (43) we denote by h the harmonic function satisfying this boundary condition and solve, as we now know how, the equation
Lv=f,
f=f-Lh
under the boundary condition v = 0 on 0. Then u = v + h satisfies condition (43) and the equation Lu = f. Consider next the boundary condition (46)
u = 0 on (,
0 E Co(9)
L. BERS AND M. SCHECHTER
240
We construct a sequence of functions 0; defined on
j
II0;
such that
- OGc'o(4) - 0
(For instance, take the 0; as polynomials.) Let us be the function in C,,. (6) determined by the conditions Lu, =f, u,
on.
By the maximum principle u; converges uniformly to a function u. Thus I'Uj - uIlc-o(rs)
(47)
0
so that u c- Co(11) and u = cb on V. On the other hand, by Theorem 3 for a compact subset 10a e G°, II u; - u, II c$+a(vo) S const. 11u, - ut 1, ca(1o)
the constant being independent of i, J. It follows from (47) that u has Holder continuous second derivatives and satisfies the
equation Lu =f. In a similar way, using the estimates in Theorem 4 and approximating the coefficients of (42) by nice functions one can
show that if L has continuous coefficients, the equation Lu =f has solutions with continuous first derivatives and generalized LQ derivatives of second order in every subdomain. 5.8.
Smoothness of Strong Solutions
We now give a proof of the theorem on continuous differentiability of strong solutions (cf. Section 3 of Chapter 1). Assume that f and the coefficients of L satisfy Holder conditions with exponent x. Let u be a function having strong derivatives up to order m in
L. for some r > 1 and suppose that Lu =f almost everywhere. We shall prove that the m-th order derivatives of u satisfy a Holder condition with exponent x and hence u is a classical solution of Lu =f. Since the theorem is local in nature, we need only to prove it in the neighborhood of some point, say the origin. Let fi(x) be a Cx function which is identically one in Ixl < 1 and vanishes for lxl > 2. Setting r7(x) = (R-'x) we have by (15a) (48)
1117 IIc,(2m = II 0C,(2)
POTENTIAL THEORETICAL APPROACH
241
for any s. For v c- Nn(2R), r > 1, set Mov = T077v
Now by Lemma B, (14) and (48), we have for v e Cm+,,(2R) II T0nv I I Cm+:c (2R)
< const. R' II v II Cm+cx (2R)
Similarly, one sees by Lemma D, that for v c- NN(R) 0(1) IIvllA-m(2R)
as R - 0. Hence we may take R sufficiently small so that (49)
Il Mov ll Cm+a (2 R) `= z II V I1 Cm+%(2R)
and
(50)
11M0vll \m(2R)
n, we see by the Sobolev-Kondrashov theorem that L1u satisfies a Holder condition
with exponent fi < 1 - (n/r). If fi > x, we have Lou E C,,(2R). Hence, by the Holder-Korn-Lichtenstcin-Giraud inequality S0Lou is in Cm+ (2R) . But by (49), (1 - M,)-1 maps Cm+.x(2R) into itself and hence Ou is in Since 0 = I in Ixl < R1, we have u E C,,,+2(R1) and our assertion is proved. If j3 < x we know that Lou is in CC(2R) and hence SOL6u is in C,,,_g(2R). By taking
-
R sufficiently small we know that (1 M0) -1 maps C,.+,, (2R) into itself. Hence u r= Cml,3(R,). But this implies that Liu is in
L. BERS AND M. SCHECHTER
242
for any
C,,(R1). Hence by our reasoning above, u E
R2 n, the argument above then completes the proof. Otherwise, we note that L1u E Lr2(R1), where r2 = nr1(n - rl) -1 = nr(n - 2r). From this it follows that u e N,., (R2)
for R2 < R1. If k is the largest integer such that kr < n, we repeat the procedure k times to obtain u e NN(Rk), with rk = nr(n kr) > n. We then know how to proceed to the final result.
-
Appendix I. Proofs of the Fundamental Inequalities
We shall outline proofs of the three sets of inequalities stated in Section 3 of Chapter 5. 1. Sobolev's Inequalities.
The proof of (a) is trivial. In fact, by
Holder's inequality Ir--A
*
('
If Ix -y1-'o(y) dy
-yl-aq dy)
Ix
I
IyI 1 f- f(t) dt q dT S const. f 00 1 f(t)IQ dt J-oo j_ t-T -oo °°
I
Clearly, Riesz's theorem is just the Calderon-Zygmund inequality
for n=1. Assume first that the singular kernel K(x) in n space is odd:
K(-x) = -K(x). We carry out the integration in the integral defining K * 0 first along the radii and then along the surface of the sphere. Thus
K*0=
fo(y)K(x -y) dy =
fo(x - z)K(z) dz
= lim f _ico() dSS f O(x 00
rn-1 dr
eI
where r = IzI. In view of the oddness of K(z) [and hence of co(z)] the integrals along two oppositely directed radii can be combined into a single integral. Hence K*
=
and
I K* l a dx < const.
f
00
aj (E) dSf
f;i=1 dSJ J
O(x - r)r-1 dr 00
f(x - r) r'-1 dr
I
dx
Since r-1 is a one-dimensional singular kernel, we have, by Riesz's theorem, that for fixed
r
I (x + t - r )r-1 drl
J-
a
dt < const.
f I0(x + t )IQ dt
L. BERS AND M. SCHECHTER
246
Integrating along axes perpendicular to
l
gives
Q
dr
O(x -
fo.
dx < const.J I0(x) I,, dx
which, when combined with the preceding inequality proves the theorem for odd kernels.
If K(x) is a singular kernel, let us denote by K the operainto the function K * 0. We tor which takes the function consider the particular singular kernels (Riesz kernels) K,(x) = cxjxl-n-1, i = 1, 2, . . . , n and note that for an appropriate choice of the constant c, 1Q2 + ... + ten = 1 = identity. To see this we observe that K. (X) = c1
a
i = 1, ... , n
Ixl'-n' ax2
Thus by (4) ci
f 10 (Y) - 0 (x)] ax Ix -yi1-" dy
Now by partial integration we have for R1 sufficiently large,
f
i
[0(y)
- fi(x)] ax Ix -Y11- dy ,
a
JF 2, we have
f
Ci
Izll-n
aR1 (x + z) dz ax,
J
c1J IZI1-n axt - J IUIi-n a0(x axz, + u) du dz
+ z + u) du dz _ C f Izll-n lull-n a2o(x ax2 Izli-my-zll-n a2o ax+ y) dv dz
= GIJ
_c
f
F(v)
a2¢(x
+ v)
ax!
where F(v)
=f
Izi1-n Iv - zIl-n dz
Clearly F(v) is positive homogeneous of degree 2 - n and invariant with respect to rotations about the origin. Hence F(v) = const.
IvI2-n
This means that X20 ,
Therefore
a2o (x
const.
+ v)
dv
J1v12-n
f
1X?¢ = const. IvI2-n 0O(x + v) dv which equals const. 0 by (1) of Chapter 5. When n = 2 the interchange in the order of integration above
is not justified by the behavior of
Izll-n Iv - zI1-n at infinity.
However, by being slightly more careful we can arrange things so that only finite integrals are interchanged. Assume that O(x) vanishes for IxI > Ro. Then by (9) we have (10)
a
Cl ai
(r-1 * K,O)
L. BERS AND M. SCHECHTER
248
But
r-1 * 1Q, = c1 lim fIzj R-. x
lx-zl>a
M(x - z) aazz) dz =
M, (x - z) [0(z) - 0(x)] dz
Rl>Ix-zl>a
- z - (x) JIx-zh=Ri M(x - z) x' dz Ix -zI _
fix-ZI-8
M(x - z)[O(z)
- fi(x)] X,
z
Ix
zI
=
dz
L. BERS AND M. SCHECHTER
250
The first boundary integral vanishes since K(x) (and hence M(x)) is even. The second tends to zero as e -> 0. Thus it will follow that
9'0 = C,
[O(z) - O(x))M1(x - z) dz
and the proof will be complete. It therefore remains to show that M(x) is in C,,, for x 0 0. This is immediate if one writes (12) in the form M(x)
=fzj< (Ix -
IR,
Z11-n
- Ixll-n)K(z) dz
Iyil-n K(x -y) dy
z-vl>R,
where R1 is some fixed positive constant less than I xJf2. The function Ix - zIl-n is in C" for IzI < R1 and vanishes for z = 0. Hence we may differentiate under the integral sign. Since K(x -y) is in C., for Ix -yi > R1, the same is clearly true Ixll-n
for the second integral as well. This completes the proof. It is obvious from our proof how one may relax considerably the smoothness assumption on &)(x).
Appendix II.
Proofs of the Interpolation Lemmas
We now give the proofs of Lemmas 1 and 3 of Section 5.2..
Assume that 9' is contained in some cube with sides of length 2R. Since we may take Ou to have compact Proof of Lemma 1.
support, we may assume that 9' is the cube. Thus it clearly suffices to prove IIOIlc,((l) < M
L. BERS AND M. SCHECHTER
264
This estimate is stronger than the Schauder estimate of the preceding chapter since only the size but not the continuity properties of the coefficients are involved.
The proof of the estimate is based on the representation theorem and on a known theorem in function theory, Privalofj's theorem (proved in the Appendix), which states that iff (z), IzI < 1 is analytic and Ref (z) is continuous on I z I = 1 and satisfies there a Holder condition with exponent 6 < 1 and constant K, then f (z) satisfies on Izi < 1 a Holder condition with exponent 6 and constant CK, where C depends only on 6. It was noted above that the function w(z) satisfies equation (6),
i.e., an equation of the form (8) where the complex-valued coefficients u(z), . . . , y(z) are measurable functions satisfying the inequalities (9). The constants k, k', k" depend only on the uniform ellipticity of (5). Applying the representation theorem we have w(z) = e81zf[x(z)] + so(z)
(20)
Here C = x(z) is a liomcomorphism of the closed unit disc onto itself which leaves the points 0, 1 fixed; the functions s(z), so(z) are continuous in the closed unit disc, real on the unit circle, and vanish at z = 1, and the function f is regular analytic for Cl < 1 and is continuous on the unit circle. Also, we know bounds and Holder conditions for the functions s(z), so(z), x(z) as well as for the inverse homeomorphism z which depend only on the constants k, k', k". I
Since T'(e'0) is the tangential derivative of a single-valued
function it has mean value zero and hence vanishes somewhere on the unit circle, so that by hypothesis (21)
1 -r' l < 2K1,
Ki depending on the Holder continuity of T'
Also, from (17) we infer, by considering the variation of 0 on a curve of steepest descent, that there exists a point zl in the unit disc such that (22)
l ic(zi) I < K21
K. depending only on the Holder continuity of T'
FUNCTION THEORETICAI. APPROACH
265
The boundary condition (17a) may be written in the form
Re [izw(z)] = -r'(z),
(23)
IzI = 1
or, since s and so are real on Iz1 = 1, (24)
Rc {izf[x(z)]} = e-812>[T'(z) - so(z) Re (iz)],
We have that x-1 (ei°) = e;n(o)
IzI = 1
0_ 1, then c-1
Proof.
Isc
1, µ1(z) = µ(z) for IzI < 1. Then 121 < kl(k) < 1 and by Lemma 13 together with (25), 2 vanishes in the disc I zl < 1 /c. It is sufficient to prove (26) for w"' and w2. Also, because the spherical distance is invariant under inversion, we may replace wz by w'(z) = 1/wA(1/z), where v(z) = A(1/z)z2/Z2 (see Lemma 10). Thus we need to prove (26) only for the case that µ = 0 for IzI > R; c and a may depend on R. Proof.
We know that If(l) I > c-1 (see the proof of Lemma 13). It follows by (23, 13) and Lemma 3 that Iw"(z1) - w"(z2)I < c 1z1 -
2211-2/p
This implies (26) if, for instance, Iz11, Iz21 0 is the measure of the point t. In the latter case we say that It is discrete. We shall denote by LZ(R",y)
the Hilbert space of all Borel functions f from R" to the complex numbers with finite square integral (1)
1112
=f If (1)21 du(t)
and the scalar product (2)
(f,g) = ff(t) g(t) du (t)
EIGENFUNCTION EXPANSIONS
307
Strictly speaking, we get a Hilbert space only if we identify two functions f and g when
If --gI =0 or, which is the same thing, when f (t) = g(t) except in a set of ,u-measure zero. More generally, let v be a dimension, i.e., a Borel function from
R" to the integers 0, 1, 2,
... and ac and consider vector-valued
Borel functions
k = 1, ... ) v(t) where the number of components is v(t) and varies from point to point. If we put
f (t) = (fx(t)),
f(t)
g(t) _ jJJ(t)gx(t) If(t)12
(3)
and
= f(t) '.f(t)
the integrals (1) and (2) have a sense for our new functions and we denote the corresponding Hilbert space by (4)
L2(R",#,v)
In the sense of von Neumann [18.3], it is a direct integral with
respect to y of a collection of Hilbert sequence spaces H, of dimensions v(t). The formula (3) means that we have referred each Ht to an orthonormal basis. This is done only for simplicity. In the last section we shall make use of other bases. We can now state the spectral theorem for a finite number of self-adjoint, bounded, and commuting operators
A,,...,Am on a separable Hilbert space H. It says that they can be simultaneously diagonalized by a suitable unitary' mapping U from H to a suitable
H, = L2(R",,,v) There are many such diagonalizing transformations U and we
can, if we want to, even prescribe that n = 1 and v = 1 (see [18.4]). It is, however, customary to choose a canonical diagonalization by requiring that m = n and that UAkU* is multiplication by the kth coordinate
L. GARDING
308
This means, in particular, that
now gives us the multiplicity
of the composite eigenvalue rr
rr
of the operators A. For canonical diagonalizations there is a uniqueness theorem (see [18.3]) : the measure y and the dimension function v are determined up to equivalence in the sense that this term is used in measure theory.
The spectral theorem extends to nonbounded self-adjoint operators A if we take commutativity in the sense that the bounded operators (A1 ± 1)-r, ... , (A. ± i)-1
commute with each other. Let us now consider some examples. First of all, we assume that the measure ,u is discrete. Then we have complete analogy with the finite-dimensional case: the diagonalizing transformation U can be expressed in terms of a complete set of mutually orthogonal eigenfunctions of the operators A. In fact, let y be concentrated to a countable set Q and construct for every in Q a set of functions *)5
k = 1, .
. .
, v( )
which vanish except at the point E where
Then Fk(E) = (F,G($,k))
for any F in H1 so that (f,g($,k))
for any f in H, if g($,k) = U*G(E,k) is the image of G(e,k) in H. It is clear that the functions G and g form complete orthogonal sets in H, and H, respectively, and are eigenfunctions of the operators UAU* and A, respectively. Conversely, if the operators A have a complete orthogonal set of eigenfunctions, we can, as we did in the introduction, construct a diagonalizing transformation U from H to a suitable space of the form (4) with a discrete measure
ju; if we want to, we can take n = 1 and v = 1. Well-known examples are the completely continuous operators, e.g., the various classical inverses of Laplace's operator in a bounded region.
EIGENFUNCTION EXPANSIONS
309
We shall now give an example of a diagonalization with a nondiscrete measure. We choose the`Fourier integral
i:rf (x) da
(W) ( ) =J e-
(5)
where
da = dxl ... dx It is well known that U is a unitary mapping from H = L2(R",a)
x = 1x1 +
. . .
,
to
H1 =
du = (27T)-n dal
... dEn
and provides us with a canonical diagonalization of the differential operators (6)
k = 1, ... , n
Dk = a/iaxk,
They are self-adjoint if we take the domain of definition of D. to be all f such that Dj is square integrable, or, equivalently
JI(ii2d,U < ac The transformation U-1 = U* is given by (U*F) (x)
=f
dy
Observe that we can write the right side of (5) as (f,o) where 0 = eiz6 is an eigenfunction of the operators (6) which is not square integrable. We shall meet more examples of this kind of degeneracy of (0.2) in this and the next section. By way of the formula
f (x) = LT*U.f(x) =
f
dy
we get an expansion of an arbitrary element f of H in terms of the eigenfunctions 0. If p(s) = p(EU .
. .
, Sn)
is a polynomial, U diagonalizes the differential operator (7)
B = p(D,i
. . .
, D,a)
L. GARDING
310
If p is real, B is self-adjoint if restricted to functions such that
dµ < 00
We have in this case a noncanonical diagonalization. It is instructive to make it canonical. This can be done by a transformation of variables as follows. Setting aside the trivial case when p is a constant, let t be a real parameter and let St be the
-
nonsingular (n 1)-dimensional part of the level surface p($) = t. Let N be the set of numbers t, for which St is not empty and define for t c N a measure cut in St by putting dtu = dt
dmot
Consider the Hilbert space Ht = L2(St,wt) of functions on St, square integrable with respect to wt, and let k = 1, ... , v(s) be a complete orthonormal set in Ht. If n = 1, is the number of zeros of t and if n > 1, we have to put x. Put H2 = L2(R,a,v) where da = dt on N and 0 otherwise. Finally, define a mapping V from Hl to H2 by putting (8)
(VF)k(t) =f
;F(E)hk(t,E)
when t e N and VF(t) = 0 otherwise. The reader will have no trouble verifying that V is unitary and that W = VU, which is a unitary transformation from H to H2i diagonalizes B canonically. An easy example is furnished by or, in quantum mechanical terms, the nonrelativistic Hamiltonian of n free particles. Observe that the spectrum of any B of the form (7) has uniform infinite multiplicity when n > 1. From (8) we obtain an analogue of (0.2) by expressing F = Uf in terms off. Formally the result is where (9)
( 4Tf)k(t) = (f,4k(t))
Ok(t,x) =Jet
`hk(t,)
- dwt()
EIGENFUNCTION EXPANSIONS
311
If Se is compact, which it is if B is elliptic in the sense that the polynomial p has definite principal part, then S6k is an eigenfunction
of B, but it is not square integrable. When St is not compact, but the functions h have compact supports, we still get functions
0, but for arbitrary orthonormal sets this is no longer true. To get an example of this, we can, e.g., put n = 2 and p = i8E2 with a larger integer s. Parameterizing St with dcul =
we get
28 d$j
so that any function vanishing at infinity and behaving like for small values of $, is square integrable on St. For such a function, however, the right side of (9) is not defined, but it is easy to see that it makes sense as a distribution. In fact, we shall see later that the eigenfunctions of differential operators always exist as distributions. III. Generalized Eggenfunctions
We are going to study situations when the formula (0.2) degenerates without losing its sense altogether. We begin with an interesting half-way abstract case treated by Gelfand and Kostyucenko [8]. First, a few notations. We let S = S(V) be the set of all complex infinitely differentiable functions vanishing outside compact subsets of an open subset V of real n-space. The results are true
also for functions with values in a finite-dimensional complex space, but we stick to complex functions for simplicity. Derivatives
will be denoted by Dk = a/iaxk and we put
D. = Dal, ... , D«,n and denote its order by Ial = m. To these symbols we add Da = 1, of order 101 = 0 and the norms max IDJ(x)I,
1«I < m
and
IDmfI = sup IDmf(x)I,
xeV
A subset W of V is called precompact if its closure W is compact and contained in V. We let S' be the space of distributions in V
(see [21]). It consists of all linear functionals L on S with the
L. GARDING
312
following continuity property. To every precompact open subset V' of V there exists an integer m and a constant c such that
IL(f)I n
It is p - n - 1 times es continuously differentiable and it is a fundamental solution of the differentiable operator
L = L (D) = Y
DX,
17.1
p
in the sense that
fC(x -y)Lf(y) dy =f(x) for any f in S(R). Let V' and V" be open precompact subsets of V which telescope so that V' c V" c: 17" c V, let h c- S be 1 on V' and put
b(x,y) = h(x)C(x -y)h(y) Then, if f E S(V') we have (3)
f
f (x) = b(x,y)Lf(y) dy
It is clear that if we choose p large enough, then b(-,_y) E H
for every}', and an easy approximation argument will show that operator L' commutes with the integration in (3) so that (4)
fB(E,y)Lf(y)
dy
L. GARDING
314
for almost all , where (Ub(-,y))($)
This is true in fact for any bounded operator from H to Hl. It is not difficult to prove that the function B can be chosen to be a Borel function in both arguments. Since U is unitary, Ib(-,y)12 =
f
da(d)
so that
j'Ib(.,.y)I2d y=f
da(d) dy
0 such that ft(x)c2(x) dot (x) < oo
Stripped of measure-theoretical detail, the proof of this theorem is very simple. From (8) we conclude that
f
(B*f) (y) = b(y,x)f(x) do, (x)
where f c C(B). It is not difficult to see that the operator U commutes with the integration and this gives (UB*f)(A)
=f B(1.,x)f(x) d«(x)
where B(A,x) = (Ub(x, -))(A)
(12)
Dividing by h we get (10) with
4(A,x) = h-'().)B(2,x)
(13)
It is easy to prove that the components of O(2) are linearly independent (see [2]). To get (11), we combine (12) and Parseval's formula, getting
f
c2(x) = I b(x,y) I2 dx(y) = f IB(2,x) I2 dd(.)
so that multiplying by t(x) and integrating
f
ff IB(2,x)12 t(x) dx(x) da(A) = t(x)c2(x) - dx(x) < oo
By virtue of (13), this implies (11) for almost all A.
Clearly, Theorem 2 applies where A itself has the Carleman property and this makes it possible to apply it to any self-adjoint operator A on a separable Hilbert space H. In fact, if we refer
L. GARDING
318
H to a complete orthonormal set we can think of it as a space of functions L2(W,x) where W consists of all integers of arbitrary sign and y assigns the measure one to every integer. If we choose all elements g1, x e W, of the orthonormal set in D(A), then where
f c D(A)
(Af) (x) = I a(x,y)f (y), a(x,y) =
is the matrix of A. Hence, A has the Carleman property, (9) reduces to
I If(x) I IAg.I < oo we have
and
A (A,x) = (Ua(x, ))(A)
(14)
I0().,x)I2
t(x)
< Co
where now t(x) IAgsI2
0 and
+. + a L* = Dma. m
0
its adjoint with respect to the scalar product (1)
(f,g) = f fg dx
We require that L be formally self-adjoint, i.e., that L* = L. The coefficients of L cannot be arbitrary. We require that at has locally square integrable derivatives of order 1. We call attention here to an obvious gap in the existent general theory. Reference in the sequal were chosen more or less at random to illustrate subject, without considerations of priority or even best results. I.
Direct Hilbert Space Approach to Solutions of the Heat Equation
We describe in this section a proof (based on a method of Vishik') of the existence of solutions of (I-1)
-Au +
at =f(X,t)
1 M. I. Vishik, Mixed boundary problems, Dokl. Akad. A'auk SSSR, 97, 193-6 (1954). 329
A. N. MILGRAM
330
where
0=
a2
axe
+
...
+
a2
and
x = (x1, ... , x,,)
a n,
0 0 for each E > 0.
fP -Pol ? e
3 Sternberg, Math. Ann., 101, (1929).
4 I. Petrowsky, Zur ersten randwertaufgabe der warmeleitungsgleichung, Compositio Math., 1 (1935). b This definition is contained in an unpublished paper of Professor W. Fulks, Department of Applied Math., University of Minnesota. See also, J. L. Doob,
A probablity approach to the heat equation, Trans. Amer. Soc., 80 (1955).
A. N. MILGRAM
336
The solution to the boundary value problem is sought as the infimuin of supcrparabolic functions with boundary values > a given function]. In his treatment of the above problem, Petrowsky restricted himself to the case of one space dimension: a2u/ax2 = au/at, and considered a domain bounded by two curves
x=0i(t)
X = 02(t),
0 - . Such refined conditions have not been obtained for the higher dimensional case. A crude sufficient condition for the existence of a barrier at a point PO E D5 is that PO be the vertex of a cone in the complement of D along the axis of which the t coordinate of a point decreases as the point recedes from PO (see figure above).
The problems discussed in the preceding sections have one feature in common in that all seek solutions of Au = U= which assume prescribed boundary values on D the boundary of a general domain in space-time. The solution sought cannot in general achieve all prescribed boundary values. For example, if IV.
PARABOLIC EQUATIONS
337
D is a cylinder with axis parallel to the t axis, the boundary values assigned at the top of the cylinder will not influence the solution
if the solution conforms to preassigned values on the lower boundary. Thus the boundary value problems considered must be regarded as a search for solutions of Au = u, which in some sense minimizes the difference between a prescribed function f and the values assumed by u on D. The word "minimize" as used in Section III means equal on a boundary set of "greatest measure." But if minimize were to mean the solution u(p) = u(x,t) is sought
so as to minimize the integral f If(p) - u(p)12 dS taken over the boundary, a new class of problems would arise. In such a problem prescribed boundary values on the top of a cylindrical domain would have material influence on the solution. A question related to the above which might be asked is the
following: let R be a subdomain of D. Let f be a prescribed function on R the boundary of R. What boundary values should be assigned on aD so that the solution u(x,t) best approximates f
on aR ? We leave open the sense in which the words "best approximates" are used. Of the methods used in Sections I and II we can say that the first is modern, the second classical. In the direct Hilbert space approach a solution of the boundary value problem is obtained by first solving Au - au/at = f with 0 boundary values. Then using f = - Af0 - afol at, where fo is a function defined and belonging to C2 in D, the function v = u + f0 provides a solution of Av aav/at with prescribed boundary values. The listener will observe at once that this seems to limit the boundary values to those which admit an extension into the interior of D of class C2(D). This would indeed be a very serious defect in the Hilbert space approach. Fortunately, in the case of second order equations the maximum principle comes to the rescue by guaranteeing that a sequence of solutions with uniformly convergent boundary values is uniformly convergent in the interior and the limit is readily shown to be a
solution. Hence the fact that the Hilbert space method is not directly applicable to all preassigned boundary values is not of vital importance. For equations of order higher than 2 the situation is different.
A. N. MILGRAM
338
V.
Equations of Higher Order
Let D be a domain in E" x J, and a = Jti, a2, ... , xK a sequence of integers where 1 < a, < n for each v. Set lal = K $12, ... , 41K. The and D" = aK/(axai 4,2 . . . 4,K), equation
a xt D"u =
V-1 jaj:z_ 21n
"
au at
will be called parabolic if Z a,D" is a strongly elliptic operator, i.e., if there exists a constant A > 0 such that
Rt[,ja.(x,t)e] z (-1),,1+1A
for2, ... ," real and
I
IEI2"
I2 = ($1)2 + ...
($1)2
0,
and (x,t) E D. Systems of equations can also be treated; definitions will be found in the other lectures. Various methods for treating
(V-1) have been given. The most direct and comprehensive is probably the method described in Section I, suitably modified. Of course, the solution obtained is at first only known to be a weak solution, this difficulty is easily overcome. Using the fundamental
solution whose existence and regularity was established (for systems) by Eidelman6 it is easy to verify by known methods that the weak solution is regular. Other methods including Friedrich's mollifiers can be used to verify regularity of the weak solutions in the interior. However the requirement that the boundary values can be extended into the interior, i.e., assumed by a function fo
which has finite m-norm, cannot this time be eliminated by a simple limiting process. For higher order equations no maximum principle is known. Thus even for elliptic equations the general existence theory based on Hilbert space methods is inadequate
to prove, say, the well-known theorem that the biharmonic equation L2u = 0 admits a solution u in the interior of a sphere so
that u = g, au/ an =f for arbitrary continuous functions f and g on the boundary. It is likely that at least for sufficiently smooth boundaries, the method analogous to that discussed in Section II will enable us to 6 S. D. Eidelman, On the fundamental solution of parabolic systems, Mat. Sb. N.S., 38, 80 (1956).
PARABOLIC EQUATIONS
339
obtain more complete results. Recently R. Juberg (Thesis, University of Minnesota) observing that for the fundamental solution (Eidelman) similar jump relations continue to hold, derived integral equations analogous to those described in Section II. The resulting system of equations, however, are complicated,
and his results are definitive in certain special cases. It is clear that an investigation of the integral equations both for parabolic and elliptic equations in an effort to obtain a Poisson integral type representation of the solution must lead to new and deeper insight into the boundary value problem for parabolic and elliptic equations.
Index
A priori estimates, 231, 263, 286, 289 Adjoint operator, 138 Analyticity theorem, 136, 139, 207
Direct integrals, 306 Dirichiet and Neumann problems, 263, 286
Dirichiet principle, 164 Dirichiet problem. 141, 152, 153, 154, 155, 159, 190, 196, 197, 237, 263, 282, 329 for the difference equation, 157 Discontinuities, propagation of, 53 Distribution, 138 Duhamel's integral, 17, 63
Banach-Saks theorem, 181 Beltrami equation, 257, 267 Boundary conditions, 141 Bicharacteristics, 58 Bounded linear functional, 179
Calderon-Zygmund inequality, 224, 245
Cauchy and Kowalewski theorem, 46 Cauchy data, 44, 62 Cauchy problem, 18, 45 standard, 63 uniqueness of, 47 Cauchy-Riemann system, 257 Characteristic form, 43, 70, 135 Characteristic lines, 7 Characteristic matrix, 43 Characteristic surface, 21, 43 Classical solution, 137 Closed linear subspace, 178
Elastic waves, equation of, 74 Elliptic, 143 Elliptic equations, 134, 135 Elliptic operators, 143 Elliptic systems, 255 Energy integrals, 24
Coerciveness, 201
Compact support, 7 Complete, 178 Completely continuous, 183,199 Complex notations, 256 Continuity method, 238, 285
Finite differences, solution of parabolic equations, 109 stability of difference schemes, 115 Fourier transformation, 65 Fredholm pair, 142, 175 Fredholm-Riesz-Schauder theory, 183
Descent, method of, 16
Free surfaces, 43 Function spaces, 216
Differentiability theorem, 138,
Estimates "up to the boundary," 235 Finite differences, 155 Finite differences, method of, 108
see also Symmetric hyperbolic systems
Fundamental identity, 211 Fundamental solution, 214, 215, 333
139,
174 341
INDEX
342 GArding's inequality, 170, 198
Liebmann's method, 159
General boundary value problems,
Linear equations with wave operator as principal part, 31
142
Generalized derivatives, 138 Generalized eigenfunctions, 305, 311 Gradient vector, 38 Green's function, 215 Green's identity, 48, 104
Heat
conduction,
boundary initial
value problem for rectangle, 104 equation of, 94 initial value problem, 98 maximum principle for, 96, 97 smoothness of solutions, 101 uniqueness of solutions, 96, 97 Heat equation, 329 Hilbert space, 177 Hilbert space approach, 329 Hilbert space methods, 165 Holder condition, 136 Holder continuity of derivatives, 136 Holder continuous, 138 Holder- Korn-L ich tenstein-Giraud inequality, 223, 244 Holmgren, uniqueness theorem of, 47 Huygens' principle in strong form, 15 Hyperbolic equations, linear, 69 solution by plane waves, 72 standard Cauchy problem for, 63 wave equation, 1, 2
Hyperbolic systems, Green's identity for, 48 with constant coefficients, 70
see also Symmetric hyperbolic systems
Integral equation approach, 215 Integral equations, 329 Interior estimates, 232 Interior regularity, 190 Interpolation lemmas, 218, 219, 250 Laplace equation, 134, 136 Lax-Milgram lemma, 180, 199 Leray-Schauder method, 285
Lions, 202
Maximum principle, 150, 160, 262, 334
Maximum principle for heat equation, 96
Maxwell's equations, 87 Minimal surfaces, 282 Mixed boundary problem, 143 Mollifiers, 145
Morrey-Nirenberg, 210 µ-conformal mapping, 267 Multi-index, 135 Negative norms, 200 v. Neumann condition, 118 Neumann problem, 154, 283 Norm, 135 Normal boundary conditions, 142 Normal velocity, 23,61
Oblique derivative problem, 154 Ordinary differential operators, 319 Parabolic equations, 2, 94, 329
see also Finite differences and Heat conduction equation Parseval's identity, 65 Periodic distribution, 168, 169, 174 Perron's method, 329, 334 Plane waves, decomposition into, 72 Poisson equation, 134, 137 Potential gas flow, 282 Potentials, 333 Potential theory, 211
Principle of contracting mappings, 283
Privaloff's theorem, 264, 267, 279 Projection theorem, 164, 177, 179 Propagation of discontinuities, 53 Properly elliptic, 144
Quasiconformal, 258
343
INDEX Regularity at the boundary, 200
Representation theorem, 164,
179,
259
Riesz-Schauder theory, 164 Runge property, 140
Scalar product, 178 Schauder estimates, 231, 238 Schauder's fixed point theorem, 286 Schwarz inequality, 166 Second-order equations, 150 Self-adjoint operators, 303, 305 Single hyperbolic equations, 62 Singular kernel, 222 Smoothness of strong solutions, 240 Sobolev and Kondrashov, 221 Sobolev's inequality, 220, 242 Sobolev's lemma, 30, 77, 167 Spaces Ht, 165, 167 Spacelike surface, 23
Spectral theorem, 303, 304, 305, 306, 307, 308 Stability of difference schemes, 116 Strictly hyperbolic, 72 Strong derivative, 137 Strong solution, 138 Strongly elliptic, 143, 144 Sturm-Liouville operators, 319 Subspace, 178
Support of a function, 7 Symmetric hyperbolic systems, 86, 87 difference methods for, 117
Tangential operator, 215 Test functions, 137 Timelike surfaces, 23 Titchmarsh-Kodaira formula, 320
Uniformly elliptic, 150, 255 Unique continuation, 162 Unique continuation property, 140 Unique continuation theorem, 262
Uniqueness, Holmgren's theorem of, 45
Uniqueness for heat equation, 97 Uniqueness theorems, 283
Wave equation, d'Alembert's solution for, 5 characteristics, 21 characteristic cone, 21 characteristic lines, 7 characteristic surface, 21
domain of dependence, 7 Huygens' principle in strong form, 15 inhomogeneous, 17, 18, 19 initial value problem for, 6, 10 mixed problems for, 35
mixed problem for finite x interval, 9
mixed problem for semi-infinite x interval, 7 one-dimensional, 4 spacelike surfaces, 21 three-dimensional, 10 timelike surfaces, 23 two-dimensional, see Descent, method of
Waves, see Plane waves and Elastic waves
Weak derivative, 137 Weak equals strong, 144 Weak solution, 138, 198 Weak vs. strong, 137 Well-posed problems, 2
Zeros of elliptic systems, 261
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