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0 eqnillity of these two
terms nils st occnr for a finite value of 1. Let 1, be the resulting interval. We thei hive
Starting a t the end poii~t of 1, repeat this process, keeping k fixed, choosing A,, 1,, until L is covered. There are cletbrly a t most lc such
...,
iutervals
4.If we now
r
sum our estimates for
J hi
IP d r
we fitrd, with
differential equations
-+ 2r
the aitl of Holder's illequality (recall that P
If we uow let to zero, bec:ruse
16
-
w tlte first term
011
> 1 , at111we obtain (2.8),
P = 1) that 2p
the right of the preceding tends completi~rgthe proof of (2.6).
Lecture 111. The Dirichlet Problem. We co~~sider now elliptic tlifferential operators, confi~ringourselves for simplicit,y to a single equw,tiolr for one unkr~own. Let L ( z , D) be a partial differel~tiill ol~erator with co~tlplex valued coefficients, w.nd let L' be the part of highest order. L is elliptic if there are no real clraracteristics, i. e;, L'(x,fl+O,
real E + O .
I t is ettsily see11 tlrat: for more tlrat~two vari~tbles,a > 2 , ellipticity implies t11at tire order k of L is evetl. In treaitiug tile Dirichlet problem we shall assume that k = 2 918 is eve]) and hirat the operi~toris strongly elliptic, i. e. that (efter possibly mu1tiplying by a stlitable coinplex function)
The Diriolrlet problem
consist,^
of finding B solution in a domain
9of
where a/& represeuts differentiation l~orlnalto the boundary. Here f and Q,, are given functions in 9and 4 respectively. We shall describe here the Hilbert space approach to the Dirichlet problem, which is based on sotne form of the projectioll theorenl, and is related to the classical method of minimizing the Dirichlet integral, In its
L. N I ~ ~ N B ~ :C On R Gelliptic partiat 0
preseut form the existellee theory is ~naiulydue to Garding, Vislrik, Browdela and others ; we refer t l ~ ereirder to [9] aud [a]for expositions and refere~rces. This and the followi~~g lecture comprise a brief description of [9]. The 0 theory is based on a sitrgle L2 inequality. Gardi~rg's inequality, expressi~~g the positive defiuiteness of the Dirichlet integral associated with the differential operator. Since this approach to the Dirichlet problem requires cousiderable differentiability assumptions on the coefficients we shall ~ssilmefor simplicity that they are of class Om in cZ) and that the boundary d is sufficiently smooth. We shall also assume Q to be bounded. Parthermore if the Qj are sufficiently smooth we may subtract from u a furlction having the same Dirichlet data as u, so we shall consider the case where the Qj vanish (3.3)
Lu=f
The Hilbert space approach yields at first
11. ,
, .
=g=
differential equations
>
preceding lecture we see tllat e fiulction ill Hj is conti~luousif 2 j a . Functions h satisfy the bou~tdary conditions of (3.3) in a geaelxlized sense. We now formulate the GENERALIZED DIRICHLETPROBLEM : #ioen j' in H,, fwd a weak so/ution 0 u in Hm of h2c= f . Using the notation of Lecture 1 we may write the operator L in the form L = 2 Dfl ap,,Dy lei, lulsm
2,
.
0
If u is weak solution in H,,, we lnay then carry out some partial iute. gration in equation (3.4) and write it as
,
B [ u , v] is linear in u antilinear in v and satisfies, by Schwarz~iuequality
We sl~all assulne the strong ellipticity (3.2) to hold uniformly, i. e. for some positive constant c, 8.6
(- 1Irn 2
IBI lul-m
ab,, (8)tflP .>c, 1 8 l Z r n
,
8 real,
for all 3 in (D. Our main result is THEOREM:For ssl~,flcientlylarge the generalized Dirichlet proble?,, for the equation (L )': 14 =f admit8 a unique solutio?z. For the eqztntion Lzc =f we have the Wedholm alternative. The .L, estimate on which t l ~ etheorem is based is G ~ B D I N G ~INEQUALITY S : There exist constants c 0 and C such that
+
>
holds for every pl in C: ( 0 ) . This ail1 be proved in the next lecture. I t is clear from (3.5) that tbe 0 inequality extends also to functions in Hm, and it follows from (3.6) that 0 the only solution in H, of (L 0)u = 0 is = 0 . Let us now prove the theorem. Suppose first that the operator is symmetric, i. e. B [pl , p] is real. and that the coastant 0 in (3.6) vanishes - which we may achieve by consideriug L C in place of L . It follows
+
+
L. NIRICNBERG : On elliptic partiat
,
from (3.5), (3.6) (with C = 0) that B [u v] serves as an alternative scalar product in the Hilbert space $,; the norms B [u u] and 1 u 1 , are eqaivaleut. We see that the antilinear functional ( f , v ) defined for all y in in, satisfies
,
and is therefore a bounded functional. By the well known represel~tation 0 theorem there exists therefore a function u in the Hilbert space H,, such that
i~ then the solution of the Dirichlet problem, and we have proved the Brat part of the theorem with C! = C . To prove the secoud part we write the equation Lu =f in the form (L C) = Cu f or
'U
+
+
Since ( L C)-1 maps H, boundedly into k,, it is completely coutinuous in H,, by a previous remark, aud from the Riesz theory for completely continuous operators we derive the second part of the theorem. Suppose now that B [ y y] is not symmetric. If we add C ( y y ) to B so that it satisfies
,
+
,
,
then we may still rely on a generalized rep~~esentation theorem due to Lax alld Milgram. We conclude the lecture with this REPRESENTATION THEOREM : Let B ( x y) be tc jovm de$ned l o r pairs of vector x 7 y in n Hilbert space H(uorn111 which is littear in a , antilinear in y and satisjies
,
,
I/),
Suppose that for some positive cottstant c the inequality (3.8)
I
B (x 7 $1 l 2 c 11 x 112
holds for every x i s H. Then every bounded antilinear functional P ( x ) admits tile representation F ( x ) = B ( v , x) = H ( x , w ) For $xed e1ei)tents 11, ,w to11iclr art: u~iiqne.
.
diferential equations Proof: For any Bxed element v , B (v, x) is a bout~dedantilinear functional of x and therefore admits the representation
,
,
for some element y where ( )H denotes the scalar prodect in H. This defines a ~ n a p p i ~y~= g A v which is clearly liuear. L e t t i ~ ~xg = v and applying (3.8)we find that
It follows that the operator A has a bounded inverse and t l ~ a tits range is closed. Fort,hermore the v corresponding to any y is unique. To see that the rarge of A is the whole space H suppose that z is orthogo~~sl to it. Then we hr~ve B (v z)= 0 for a11 u . From (3.8) it follows, by setting v = z , that z = 0 . Thus A maps onto the elltire space, and therefore every a~ltili~learfunctional ' P ( x ) being of the form (y, x ) admits ~ the representation P(x)= B (V $2.). The other representation is proved in a similar way.
,
,
Lecture IV. A Priori Estimates. Before provil~gGardiugls inequality let us make some general remarks about a priori estimates. Consider a differential equation h =f of order k and assume that the solution has been made unique by some auxilirry conditions. One wauts to study the inverse operator - to see, for instance, to what class of functions the solution belongs, i f f belonge to rt give11 class. For this problem, and also for the existence theory, a priori ineq~u~lities play a basic role. Let us suppose that the auxiliary couditions are homogeneous, then a typical a priori estimate would assert that for some uorm 11 11
11 Dfl u 11 zz
constant 11 Lzc
11
For instance, if we kr~owthat the equatiol~has a solution of class OK for all f uf class 0 then indeed, by a simple application of the closed graph theorem, we would have
II Dfi 21 1) < constant )ILzc 11 ,
1 B I s; K -j
,
L. NIEENBEBG : 0s elliptic partial with 11 11 the usual uorm.in Cj. 111 general if Lu has fit~ite (1 11 norm we will not obtaili such an inequality for K = k , rather K < k ; that is we c a ~ ~ ~ esti~riate lot iudividu;~lly all derivatives enterir~g ill L . However I believe that elliptic equatiolis oil11 be cltari~cterizedas tllose for which oue can e s t i ~ ~ t a all t e derivatives, i. e.
for a wide class of n o r m (this is skated its it co~iviction~ i o ti~ tlieorem). Cousider ttow an elliptic equatiott Lu =f with saitwble lto~nogel~eoas bouriditry coaditions. Most i h priori estimates are just of tlie type (4.1) or, if oue does not assume uniql~el~ess, of the for111
Iudeed ~ n u c hof the theory of elliptic e q ~ ~ a t i ois~ concerned ~s witlt proving s u c l ~estilnt~tesfor various liorms 11 11 a11(1proviug attalogo~lsestimates for fuuctious with no boundary restrietioos :
,
Here a is ally compact srtbdomai~tof 9, and the norm 11 1 " is cousidered only for fu~rctio~is defined in a I A word qf oautiow: The estilrlate do not hold for the most obious norm tliat orhe would try, ~iamelytlie 111axi1n111n (or Go) norlrl lior i l l getteral for Cj nonns, however they do hold for Cj+"iorn~s, 0 a l , and for many illtegral norms. W e quote some immediate colisequence of (4.2), (4.2)'. 1. If f aud the coeffieiet~ts of L are in Cm t l ~ e na solutio~t of Lu =f is also in COO. This follows fairly easily from (4.2)'. 2. 80lutiotis of Lu = 0 with bounded llorlrr 11 11 form a compact family. This follows from (4.2)' itutl the Calculus Levma: The set 11 # (1 11 Du 11 coltstant is conlpuct in the 11 as norm. q a o e with (1 This lemma holds for a wide class of norms. 3. The set of sol~itionsof L.u = 0 sitisfyiug the bouudary conditions (so that (4.2) holds) is fiuite dimensional. This follows with the aid of the Oalculus Lemma. I would like to describe briefly a general recipe for proving such estimaten. This cousists of several steps : 1. I11 case of (4.4)' prove it fil~stfor equations with constant coeffioiente and only highest order terms, and for functions of compact support.
.
<
lyl-ns
O P , ~5P 57 2 c0
I 5 12m
for a11 real
5.
Here the notatio~lsof Lecture 3 is used. I'roof: We prove first the sufficiency, following our recipe. The Calculus Lelr~rne(4.3) will be used in the forlrr: For every E > 0 there is :I aol~stant C(E)such that for every Gm function u with compact support
Tl~isis contained in our inequalities of Lecture 2, but is most easily proved with the aid of Fourier transforms. We consider now t l ~ edifferent steps in provir~g(4.5), the Step 2 of the recipe doe8 not occur here since our fnrlctio~~s have comlract support. 1. Suppose that the oP,, are c o ~ ~ s t t tat111 ~ l t vr~irishunless IbI = l y l = m . We ilrtroduce the Fourier t,ral~sformof u
By Parseval's theorem we have
proving (4.5) for this special case. We uow consider' the verii~blecoefficie~rtcase and breitk Step 3 into two parts. 2. Suppose that the support of u is sufficieatly small, contained, say, in a small sphere about the o r i g i ~ .Then accorfing to the preceding inequality we have
diff6rsntial equations
If now the support of u is so srr~allthat cp,, hi18 smi~lloscillation there we see that the second term on tlre riglrt may be bounded by
The third term is trivially bolrnded by oo~lsta~it 11 u ( I , Thus we find that
(1 u
from which follows the inequality
(4.5) now follows with the aid of (4.7). 3. Consider )low the general case. Uonstruct e partition of unity in
with the support of each wj as small as desired. Then
by the preceding Case 2, 2 constant (1 u 1 ;
+ 0 (11
u I)m
. )Iu (In,-1)
and the desired result now follows easily with the aid of (4.7).
a,
L. NIRENBERG ': On, elliptic partial
,
We see that the constants c 0 iu (4.5) depend OII c,, a11 upper of the leadi~rg bonr~d for the cg,, 1 , i111d 011 the modulus of co~~tiunity = y I = n b , a11d finally on tlle iion~ail~ 0 . cBly with I Now for the proof of the nacessity of (4.6). Suppose that (4.5) holds aud that the left Itand side of (4.5) va~~islles for solne real I[ I , and some poiut ill (i3, say the origin. Followiug the ergument iu Step 2 in the proof of sufficiency we see that tlie inequl~lity
I I
I
[=r, I/=
llolds for all Cw u with support in sonle fixed neighborhood U about the origin and in (2). Set u = e""ra. c (a) for real I where 5 (x) is a fixed real Cm f u a c t i o ~with ~ s~rpportin U and in 9 , One sees reitdily that as I w the left hand side of (4.5)' is 0 ( P I ) and not o (IZm)while the right hand side is 0 (1%"'-I), so that (4.5)' does not hold. C Garding's inequality (4.5) is a t one end of a whole spectrim of interesting and useful ineql~alities making tliffere~ltrequirelr~entson u at the 0 bou~ldary,Gardiugls inequality niaking the maximill reqaire~nent- that a t the boundary. A t the all derivatives of u of order less than nt va~~islr other end of the spectrunl is the ir~equalityof Arouszi~ju[13] involving no boulrdary conditions whatsoever. Aro~lszaju co~lsiders a 11ulnber of dillerential operators Lj (8,D), j = 1 , N of order ~ 8 with , c:oefficients eoutinuons ill the closure of a bou~~detldomain (a, and solves the following problern : Under what conditions can one assert that for 1111 C" functio~lsu in 9 the inequality
,
-
... ,
holds, with ttle ao~lstant iudepeudeut of , u ? He gives necessary and sufficieut conditions : (a) the operibtor 2 Lj L; is elliptic, here Lf is the formal adjoint of L j . (b) A t any boundary point - 8 of 9 , if is the unit normal to ci) m d 5 . 10 is any real vector taugent to 6 then the polynomials in z , -+ Lj (x, 4 z n) have no common complex root z . Here Lj is the leadiug part of Lj An exs~nple of Aronzsaju7s illequality is the following; for functions u (x, y) in a bounded domain in the plane
+
.
J1
uW 12 ilx dy
sz constant
I
(1 urn l2 + 1 u, l2
+ I u le) dx dy .
diffevential equations Even this simple extcmple is not trivial to prove. Since the report of Aronszajn a 11nmber of people Ileve coottsidered the problem *of proving (4.5) for v n r i o ~ ~quatlrat,ia s forms (4.4) and under various differential boundary oo~lditions. For one operator L j Agmon, Douglis, Nireuberg [14], (in a fortheotning paper which will be discussed later) have characterized these dilTerential boundary coilditions whicb are m/2 in unmber aud for which (4.8) holds. Scheclrter [l5] has treated N operators a l ~ dgeneral boundary conclitions. Aronszaju, in i ~ n p ~ ~ b l i swork, l~ed has treated the geaeral problrtn (4.5). Also Horl~landerand Agmon [l6] have solved the general problem for (4.5) a,nd genelnal ciifferel~tiiclboundary conditions. The proofs follow the recipe outlitled above, the trraii~step being the first, for functions in a llalf spibce. We conclude the lect~ire with a result that will be used in proving the differe~rtiabilit~y r ~ t the boundary of solutiolls of elliptic equi~tions. Iu the followil~g ,ZR denotes the henlisphere I x I R x,, 2 0 We shall denote the varia,ble x,, by t (xi, x,,-,) by x and (xi ... x,) by (x, t) . Lemma: Let u be a weak solutio~tof a diflr.etttia1 equatios (of order k) with, for simnplioity, CW coeflcienfs,
,
< ,
... ,
, ,
.
i n the interior of a hemisphere Z R , where .b are giver&f~cnctions,mid assultre that the plane t = 0 i s ~iowhevecharacteristic, ill jirct that the coeflcient a of I$ i n L does not vtrlzish. I f fol* every 6 0 the funelions .fP,IIb lc fbr I /?1 <j , D, Dj w belong to Lz i r ~2R-sthen trlro the ftrt~ctioftDft' 11 hcts this property. For j 2 k - I there is notl~ing to prove, as we may solve for the functioll ~ ( " z c froin the differential equatiou (4.9) operated on by D::+'-~. Thus we suppose j k -1 . The proof ltlakes use of a well known formula giving explicitly a smooth extension of a function v defined ill a half spitce t > O to a fu~~ction defined in the full space:
>
.
+
,
,
where the us,, b e l o ~ ~ so g L,, autl that derivatives D, 1)jv and v itself belorrg to L, ., For N sufficierltly large we now exterrd tire filrrctiolls v , v,,, to negative t defining UN by (4.10)a l ~ dv , , , , ~by
,
One may then verify that the equation
holds in the entire 8pace iu the weak sense, R I I ~that tire v , , , , ~ the , derivatives D, DjvN and V N itself belong to L 2 . Let us now take Folirier tr:rr~sforr~~s with respect to 8 and t , and write ( E l , E,,-,) = 5, En = t Denotirrg the trsasfonn of a function f by we find that
...,
7
w
.
GN a r ~ d1 5 1 (1 E I j f I t I j )
,
belougirrg to L, in the (E t ) spice. To conclnde the proof we have to show that belougs to L 2 . To this end write
with
VS,,,~,
differential equations We shall show tJhat ertch term on the right belongs to L,. Pron~(431) we find that the first tern1 on the right is bout~tled by
Since s $ I y I < k -j - 1 it follows t h ~ the t fact,or of v , , ,is~ uniformly bounded, and hence t h i ~ tthis term belongs to I,,, since the V,,~,N do. The second term on the right of (4.12) is bounded by
with o an absolafe constant, and hence belongs also to L , , by an earlier remark. This completes the proof of the Le~n~na.
Lecture V. The 1)ilferentiabilitj- of Weak Solutions of Elliptic Equations
In this and the next lecture we slrall preseut a self contained proof of the well known result thiht solutions of elliptic eqllations with C m coefficients are of class Cm. Many proofs exist in the literatoi~eincluding proofs for Inore general 1171, Malgrange [18]. The proof here classes of equations, see Hiirn~;u~~ler seems rather straigtforward; it i~ based essent.ially OII a proof given by Lax [19] and is closely related to proofs give11 in lectures by Bers [20] a ~ l dSchwartz [21] (see also [9]). We confine oiirselves as before t,o a single equation (not l~ecessarily strongly elliptic) although the argument extends also to systems. Dieretitiability !L'heorenz : If u is a locally epucrre ittteq~ableweak sohtion of the elliptic epuatiofo L u =f, alzd f E Cm tlbelz u E Om. Remark : If u is a distribution solution theu u = Ak v for some continuous v (here A is the Laplace operator), and v is theu a weak solution of L A h =f . The Theorem holds therefore for this case also. The proof consists in showing that u has L, derivatives of all orders in every compact subdomain. That u E GM" tl~en follows from the Sobolev estimates proved in Lecture 2. However since we ollly need a very simple case of the Sobolev lemmas me present a separate proof of it here.
Lemzcin (Sobolev): In n u ~vnoothu dot~aaitt9 i f u hrrs I;, derivtctives up to order s in for s 9812, thes u is continttous i n 0 . In fact
a
>
Proqf: The first assertion follows easily from tile iaequality. To prove the i~lequalily let x, be an inner point in 0 (for 8irlrplicif.y take x, = 0) alld suppose tlrere is 'a sphere about x,, in 0 with radius R . Let furthermorl 5 ( r ) be a function iir Cm, e q ~ l ~tol 1 for 0 T L R I 2 , and vanishing for r 2 R By ii~tegratiou alolrg ally rl~tliusfrom zo= 0 , aud by repeated partial integration we see that
using ScLwarz i~leqnality.For. s n/2 the last iutegral is finite. If the boundary of 0 is such that a t any point in there exists a cone with a fixed ope~iiogalrd length contailled in then tire same proof holds; instead of illtegrating over the full sphere of radial directions, we merely integrate over the directions lying in the cone. The proof of the Differentiabilih Theorem co~lsistsmainly of a series of simple lemmas of c a l c u l ~ ~coucer~~ed s with a special situatioa) that of periodic functious, and tlris lecture will confirled to these oalculus statements. We consider peviodic fulrctio~rsu E C* with period 2 n in each z,. For such f u ~ ~ c t i o utire s Fourier series ,u=XuEei..f, 5
( f j = in tegor) converges uuifornll y.
E = (E, . ... .5,,)
By Parseval's equality me have the following estimate for each nonnegative 8 (5.1) ,
+1 E
canstant 2 (1 5
+ I iI?
I - constant 2 (1 C
)S( U E I'
where the i ~ ~ t e g r aisl taken over :L period cube. For any integer 8 we introdl~cethe followiag scalar prodnct and norm, differing slightsly from our previous notation,
,
,
We write (u u), = (u u ) and proceed with the Calculus : 1. 11 u Il is inocretisillg in 8 Flirlhern~ore for t, E 0 there is a constant C (6) such that
.
>
< < t, 8
and any
< +
Proof: For any a 2 0 , as E ot9 C ( 8 ) ot1. 2. Set p , = ( l - A ) t u , y = ( l - A ) t v , sethatp,=2ue(L+I&12)teb.~. e From this we bud
As a consequence we have Lemma : If o E Cm, then (5.5)
Proof: find
,v)t = (a W v)t 4- 0 (I1 llt I1 v + I/ Assunie t < 0 . Using (5.4), (5.3), (5.1), and (W 11
1
91
]It-1
Ilt-I
partial integration, we
( w u , ~ ) ~ = ((1 w- A ) - t p , ! ~ ) = ( ( l - A)-tp,
Iu the case t / 0 the proof is similar.
I1 v (It).
zy)
L. NIRENBERG : On elliptic partial 3. Sckwar(t)t'a inequality : (Clear I)
Proof: According to (5.6) tile left side of (5.7) is 11ot smaller than the right side. If however we set v = (1- A ) t t z , then, by (5.4)
proving (5.7). We can now form Hilbert espace Hs by colnyletil~g Cm fullotions in 11,. For a > 0 these agree wit11 our previous definitious. the norms Obviously H,c Ht f6r a t All the previous results llold for fuuctions with the appropriate i~orlnsfinite, for insti~~lce (5.7). We lnay regard H, as give11 by a forlnal Fourier series with fiuite 11 11, noun. We remark that the scalar product
11
>
.
,
,
is defined, by extension, for any fiu~ctions11 E Hg v E H-, and that ally bounded linear fu~~ctional f (u)defined on Hs lnay be represented in the form
with v E H-,; this follows imlnediately from the Fourier series representation, so that we ]nay regclrd H-, as dual to H s . Though me shell not use this, we reulark thirt the closed ulrit ball 11 u 11, 1 in H, is compact, i l l Ht for s t We continue with the calculus. 4. Consider any differential operator L of order k with Cm coefficients.
.
Claim :
More precisely
, ,
where c = G (k u) K is a bound for the leading coefficieuts, and K' is a bound for all coefficients and their derivatives u p to order 1s 1.
P ~ . o o f :S i ~ ~ cobviously e 11 Di 24,II 5 C O I I S ~11 14 prove (5.9), to show that i f a E Ca theu
1
it soffices, in order to
I
where k' a11t1k" are bounds for a 1 aud 1 Dj (1 (j 1 s 1 ) respectively. P ~ o o fof (5.10) : Co~~siderfirst the cave s 0 . Set 97 = (1- A)' ly' = (1 A ) S a11 theu we I~iive,by (5.4), and partial integratiou,
-
lntegratiug t l ~ el ~ , s by t pt~rts(-
So dividiug by
11 y
8)
< ,
where o depends only os P and n .
where c is an absolute oonstnst. Here (f)-112is defined in tavms of the Fourier transforvn o f f by
T(E)
There is at1 L, analog~ie of 3, which is however more complicated to state. We call Part 1 of the theorem a result of Privaloff type. I t is a simple extension of classictal results of Holder, Giraud and others, to wllicl~ it reduces if we set t = 0 . Part 2, a result of Riesz type, is a stnlighforwlttd extension of recent results of Ualtlero~~ a1111Zygmund [24], to which it reduces if we set t = 0 . For the special case of the HiIbert transfonn for ,n= 2 it is due to Riesz, a ~ ill ~ dfact it is proved by reduction to the Riesz result witlr the aid of a, device of [24]. Part 3, is proved with the aid of Fourier transfor~ns- one sl~owsthat the Fourier tri~llsfornl K ( 5 ,t) of K ( x ,t) with respect to the a variables is bou~~ded i l l absolnte value by constant (1 t I)-' from which the result follows easily. Part 3 plays an essential role in the derivation of the L, estimates. A
+1
,
Lecture VIII. The Boundary Value Problem in a Half Space; The P o i s s o ~Kernels. ~ I n this lecture we shell show how to solve explicity the elliptic system (7.1) with coasbant coefficiei~tsfor the special case of a half space. Making n slight change of llotation we sb@ll co~18iderthe space to be
L. Nrnlc~slcaa: On elliptic parNal
+
, ...,
n 1 itimeusional, mitli tlie first n coordinates denoted by a = (3, x,~) and tlle last coordinate by t . 111 the half spikce t 0 me cousider-for sim-
>
a
plicity the homoge~ieousequation, with D
where L is an elliptic operator of order 2 m with only highest order terms, satisfying the e coutlition 011 L B of the ~reviouslecture, i. e. for fixed real 5 =(l, tn) 0 the polyuolnial L ( E l z) lias exactly nb roots z on each side of the real axis. 011t = 0 we prescribe the derivatives
,... , +
I$-1,'= Gj(x) with the Qii in G r , for simplicity. Tlie solution w'ill be given i l l terms of kerliels Kj ( a , t ) j = l the Poisson kernels,
,
,...,m,
Kj(~-y,t)Qij(y)dy=2Kj*Qij,
(8.3)
where (: deuotes co~ivolution. Our constrrlction of the Ki is an exteusion of the coustructio~lgiveu by Agnion [25] in two dilneasions, n = 1 , but it is based OIL the Fritz Jehu ideutity (1.6) of Lecture 1 : For QJ (a) in Go"
where q is a non-negative integer of the stluie parity as s, A is the Laplacean, ant1 the priucipitl bnruch of the l~garithm is taken with the plane slit along the negative real exis. First some preliininaries. For fixed real t 0 denote by z t = z$ (6) k =1 m , the roots z with positive ilnaginary parts of L ( E l z) = 0 , and set
+
,...,
,
."
at
+
The coefficieuts are an~lyt,icill 6 for real 8 0 , a l ~ dIio~~iogeneous of degree p . With M+ we associate the polynomials (in z)
diflerential equations
The following relations are easily verified.
where y is a rectifiable Jordan coutour in the complex a plane enclosing all the roots r+(5) ill its interior; 8; is the Kroi~eckerdelta. We call uow writhe dowll the Poisson Kernels : For j - 1 2 a
for j-1 0 eiiclosiug ell the roots z of M + ( [ , a ) for all 1 61 = 1, l real. Before proving thiit these for~riulwsrepresent Poisso~~ kernels we observe, with the aid of the identities
(-
1y+r 2"~
(p+A)!(-l-p)!(2Y(
3
log -r = z p , pO
for 1,zc integers, and ~ 1 , ~sott1e ' apl)rnl)riate co~~stants, that sr may rel)rese~rt the fal~ctioi~s K, iir the forn~- with q a I I O I I - I I ~ ~ ~ integer L ~ ~ V ~ 11avi11g the same parity as n. -
L. NIBENBERG : On elliptic partial
44
where, for j - 1 2 n
,
and for j - l < n
+
I t is easily seen that Kj,, and all its derivatives up to order j q are contiuuous in the closed half space t 2 0 . We uom prove that the kel.aels Kj give11 by (8.7), (8.7)' are indeed Poissorr kernels. By iuspectior~ me see t l ~ a tthe Kj are analytic solutious of Lu = 0 for t 0 . H e l m a defined by (8.3) is ib solution. Setting
>
we shall show that uj beloltgs to Cw iu t 2 0 alrd that for t=O, k=1, Col~sitleriury pa,rt,ial derivative of ortler s of tcj q of the same parity as n , altd such that q 2 s -j
.
...,m.
Clloosing au iuteger
+ 2 we have, for t>O
after partial integration, recalli~igthat !Dj€ G.Since, as remarked iibove, D-j,, is continnous in the closed half space t 2 0 it follows that DSuj c m be exte~rdedas ;G contilrtious fiil~ctionill the entire closed l~alfspace t > O . Since s is arbitra,ry we have proved that uj E Cm iri t > 0 .
To verify (8.1 1) choose q snfficiently large so t l ~ a tq >j j =1 m Usillg (8.12) we have, for t = 0 ,
,..., .
after a change of variable. Assume first that k * j appropriate constants c', c"
. Usirlg (8.10)',
- 1c
+1 .
(8.10)" we find, for t = 0 , and
+.
by (8.6). Thus (8.11) is proved for k j Now suppose k =j If j - 1 n we have, usil~g(8.9) (8.10)' arid (8.6)) for some constant C'
.
- pj (jan i q!
>
eSdW6 I(y .E)q (log 7+ c/)/% ICI-1
M+
rj-l
Y
.
where yq (y) is a hornogeueous polynot~tislof degree q Similarly if j - 1 n we firlil, usirlg (8.10)" ilncl (8.6)
1). ~Yupposcthat there exists a constant C such that for every subdomain G,; G, c G :
the Diriohlst problettb
for all suficently small vectors h. Tile% $6 E Hj+l,Lp(G) and
Proof: Consider first the case j = 0 . From (3.3) and t,he weak compactness of the unit sphere in I;, it follows that there exists a sequence of vectors (h7m)z-,in the direction of the ~6 axis, hm 0 , such that the sequence dnm u (wt sufficiently large) tends weakly in Lp (GI) to a function ui; and this in every fixed subdomain G, G, c G. Since 11 ui ( C for all such subdomains, it follows further that ui € Lp (17). Now, from the definition of weak convergence we find that for all fun-, ctions g, E C r (G):
-
I~L~(G,)
,
S
g, ui dx = lim m-m
G
S
g, .
u dx
G
m+w
This shows that ui is the distribution derivative Di u in G. Since Diu € l;,(G) (i = 1 n) me conclude from Theorem 3.1 that u E (C). Clearly, we also have I1 Di u IIL~(G! I c .
, ...,
Next, assume that j 2 1. Let again (h"] be a sequence of vectors in the direction of xi tending to zero. I t is easily seen that a,, u converges to Di u in Lp (G,). Assuming without loss of generality that G, is of class COJ and applying Lemma 3.1 to the sequence Id,,, u ] ,it follows that Diu E Hj,Lp(G,) and that 11 Di t~ l l j , ~ p ~5~ , c~ (i = 1 , n )
,
From this and from Lemma 3.2 we conclude that u E Hj+l,Lp(GI) for any subdomain Gi of class CoJ (and consequently for any subdomain G, G, C G). Since all the distribution derivatives of u of order (j 1 are functions belonging to Lp (G) it follows from Theorem 3.1 that u E Hj+l,Lp(G). That (3.4) holds is obvious. By the same argument used to prove Lemma 3.3 for j = 0 one obtains R, x, 0 Let t~ LEXMA3.3'. Denote by ZR the hemisphere I a ( be a function beionging to Lp (ZR),p 1. #uppose that there e ~ i s t sa coastant C such that for every R' R:
+
.
S H M U ~AGMON I. : The Jp approicch to
,...
for all suficiently small vectors h of the form h = (hi , h,-, ,0). Then the n - 1 are functions belonging to distribution derivatives Diu for i = 1 , L p (ZR)with 11 Di u 5 0. The following known lemma will bc useful. LEMMA3.4. Suppose that G has the cone property. Then, for all functions u E Hj,Lp( G )( j 2 1) and every e > 0 the follou~ing inequality holds :
...,
,
where C is a constant depeuding only on e ,j p and G. Lemma 3.4 for somewhat more regular domains was established by Nirenberg [24](5). The inequality for domains which have t,he cone property was proved by Gagliardo [13]. Finally, we conclude this section with the well known integral inequalities of Sobolev [30]. THEOREM3.2. Suppose that G has the cone property. Then the functions u belonging to ( p > 1) satisfy the following relations. n 1 1 then uE L, ( G ) where q is de$ned by - = - - j Also, (i) If p < 9 q p n
.
with a constant depending only on n ,j ,p and Q. 'n
(ii) If p == : then u EL, (G) for every 1 9
(iii) If p
n >then u is a continuous function
null set) such that (3.6)'
< q < co and
j
SUP I u I 5 Const. a
11 u I(~,L~(G)
(3.6) holds.
(after correction on a
7
with the same constant dependence as above. REMARK:If the boundary of the domain is somewhat more regular, e. g. if G is of class 0081, one can assert in. case (iii) of the theorem that u satisfies a Holder condition in Cf. 4. Some lemmas related to elliptic operators with eonstat~tcoefficients.
Let A ( x , D) be a linear differential operator with complex coefficients operating on functions u (x) defined in a domain of En;,. Denote by A' the
(3)
The analogous one dimensional case is due to Halperin and Pitt.
the Dirichlet problem
leading part of A, i. e. the part of highest order terms. A is said to be elliptic in the domain if for every point x in the domain the characteristic form A' (x 8) 0 for all real vectors 8 = (E, 5,) =!= 0. I t is well known that if n 3 and A is elliptic then its order is even. This is not necessarily true for n = 2. In this section we shall consider an elliptic operator A of even order 2m with constant coefficients and with no lower order terms :
, +
, ... ,
>
A ( D ) = 2 a , Da lal-2m
A being elliptic there exists a constant 1
.
> 1 such that
for all real vectors 8. We term 1 the ellipticity constant of A . We denote by xf= (xi , x , - ~ ) the generic point in En-,and whenever convenient write x in the form (a', x,). We also put, D,,= (Ill Dm-,) and D = ( D x f D,) Write the operator (4.1) in the form A (D,, , D,). For a fixed real vector E' = (5, , En+) 0 consider the roots (in &J of the polynomial d (5', 5 , ) . If n 3 the ellipticity of A implies the exactly half the roots possess a positive imaginary part (see [3]). This is not necessarily true for n = 2 if the coefficients are not real. In general we shall say that A satisfies the e roots condition H if for every fixed real vector 5' 0 the polynomial d fn) has exactly m roots with a positive imaginary part. The following two lemmas are basic for the proof of regularity in Lp of weak solutions of elliptic equations. The 6rst rather known lemma will be used to establish interior regularity (and Lp estimates) of weak soliltions of elliptic equations and overdetermined elliptic systems. The second lemma will be used to establish regularity at the boundary of weak solutions of the Dirichlet problem. In both lemmas A will stand for the elliptic operator (4.1) and p will denote a number 1. In Lemma 4.2 we shall assume in addition, if n = 2 , that A satisfies the e roots condition ,> introduced above. We shall denote by SE the sphere I x R and by 2Bthe half sphere Ixl O . LEXNA 4.1. Given a function f E C r (SR) there e ~ i s t sa function vE Cm (&) such that
... ,
, ... ,
, . ..., +
>
+
(r,
>
I
1 , consider
for so?)be constant G. For v € ELp(E:) p
the transform
Then, u. 6 Lp (E:) and
where y is a constant depending only on n and p. Proof: Set \dl(x)= for x n > O ,
I M ( x ) = - ~ x l - ~for
Extend v as zero for.
xn< 0 .
.
c, 0 :
8%
-
Now, M ( x ) is an odd homogeneous kernel of degree n bounded on I x 1 = 1. Eence, we are in a position to apply the Calderon-Zygmund theorem [8] to the last integral (4.17), from which it follows readily that
y depending only on n and p. This proves the sublemma.
To prove Lemma 4.3 we shall first transform formula (4.11)'. To this end note that (integrating by parts with respect to yn)
i
0:-'w (y', 0 ) .Kj (x' - y'
En-1
,Xn)
dy'
the Diriohlet problem
where here and in the following all differential operators under the integral sign act on the y variable unless otherwise indicated by a subscript. Summing (4.18) over j = 1 m me obtain for the solution u of (4.11) the representation :
, ... ,
where (4.20)
Using (4.19) and (4.9) we observe that if q is a non-negative integer having the same parity as w - 1 then
where
ZjBq are
kernels given by
From (4.20)' it is readily seen that the inequalities (4.10)-(4.10)', satisfied by the kernels K,,, , are also satisfied by the kernels Put :
&,, .
1n
so that by (4.19) u = 2 u j . To establish the lemma it will suffice to show 0
that the inequalities (4.12) and (4.12)' hold for u j . We shall prove this for j odd. The proof for j even is similar.
SHMUICI. AGMON: The Lp approach to
+ + .
Choose q = 21th n 1 From (4.22) and (4.21) we obtain after obvious integration by parts with respect to y' :
Differentiating (4.23) ye thus obtain :
Suppose, firat, that I u I = 2 m . Using the estimates (4.10)-(4.10)' which, as was pointed out before, are also satisfied by the kernels Zj,, (q=2mf n+l), we find that
with a constant G depending only on n , nb and A. Thus, applying the Sublemma to a typical integral of (4.23)' it follows readily that
where y, depends only on n and p. This yields (4.12). Suppose, now, that 0 (( u 1 2m - 1 Prom (4.10)-(4.10)' one finds readily that in this case
.
I a A,,
1 . 7 (3-1)
6, (x', zn) I < Const. ( I x
Da
(-(n-l)
+1 x
1Zm)
the DirieAlet problent
,
with a constant depending only on n nz and 1. If, furthermore, the support of w is contained in it follows easily from (4.23) and (4.25) that for IuI 2nb:
zB
,
,
a, E C lal+j-2m(8) for
1 a 1 > 2m -j ,
tokerea,s the remaining coefjcients are measurable boufided functions in G . (ii) The fol2occ;ing inequalities hold in ($ : and
I D P a a l ~ Kfor
lul>2m-j,
la,l 1 and
C i s a constant. 1
+1
Denote by p the exponent conjugate to p' : - - = 1 P P' u E H& ( S R ) . Moreover, jf 0 R' R and R, = ( R R')/2 then
+
<
p' it follows from (6.1) that we also have
.
.
,
for all functions cp E fl (BE). Hence, by the result just established ( p replaced by q) we conclude that u E H : ; ~ ( & ) .Invoking Sobolevls inequalities (Theorem 3.2) it follows that u E (SR) if either q 2 n or q n but 4, = q n / ( n - q) > p . On the other hand if q n and q, < p ~ o b o l e v ~ s inequalities give only that u E L$ (SR). In this case (noting that q, q) we repeat the same argument with q replaced by q, ; either arriving at the desired result zc E L ? (Ha) or at lea.st proving that u E L:' (SR)with q, q, Carrying on in this manner we obtain after a finite number of steps that u (SR) This yields the theorem for j = 1 To prove the theorem for j 2 we use induction - supposing the theorem is true for j - 1 ( 1 <j - 1 2 m) we shall prove it for j . We first observe that without loss of generality we may assume that A contains no terms of order (2 W E -f :
LP'
.
ELF .
.
0 , define:
g , (x) = 9 (x - E) for
1
,
5 6 ZR
for ~
g,, (x) =.O
8 2.
1
€ x
< ,
'+l
u(a,gn:,)=2 j=l
x' = (xi,
4~
.
( ,--7) for XI
xn
-
Go c G,
LP
>
, ,
.
c , c G , then
,
where c is a constant depending only on n , max m i , p , N , K , the ellipticity constant 1 and the domains. Proof: Put nto = min m i , m = max m i , and let d be the distance bet-
xi
.
ween dGo and dG, Denote by the differential operator with coefficients complex conjugate to those of Ai. Given a point x0 E define :
6,
where A is the Laplacean. A& is a linear diiferential operator of order 2nt with coefficients satisfying Condition [ I ;co K ] in co being some consta,nt depending only on w nt and N . Also, A,o is elliptic at xOand conseqnently, by continuity, is elliptic in some neighborhood of xO. More precisely, since the coefficients of the leading part AAo possess first derivatives bounded hy c O K , it is readily seen that there exists a positive number Q < d , Q depencling only on n m N , K , 1 and d , such that
a,
, ,
, ,
1 .
Suppose that for all functions v E GZrn(G ; {D")l,ls,n-l) the .following inequality hoWs :
I (U ,AV)GI 5 C 11 v 112m-j,~,,(~)
(8.3)
where A is the elliptic operatw (8.1), j is a positive integer 5 gm, p' > 1 and G a constant. Suppose also that the coeflicients 0.f A satisfy condition
where c, is a constant depending only on n , m , p ,E,1 (the ellipticity constant), and the domain. Proof: By an obvious covering argument it suffices to show that for every x0 E there exists a neighborhood Q' in the relative topology of G such that u E (Go), and such that 11 u I l j , L p( u j is majorized by the right side of (8.4) with a constant c, depending in addition on f P . For a point x0 in the interior this follows from Theorem 7.1, taking for Q0 a sufficiently small sphere with center at xO. Suppose that x0 E 8 G In this case there exists a sufficiently small neighborhood G of xO in 6 , and a measure preserving homeomorphism ( 9 ) of class CZm : I which takes 5 onto the hemisphere :1 / 1 )0 Let A be the transformed elliptic operator under the mapping and put Let, = u (x (G) and defined in further, be an arbitrary function belonging to C2n" (Zi; (Da)lalsm-l)and vanishing in some neighborhood of d,2, (the curved part of 82,). Put v (z)= Y(g(2))and extend v as zero in I t is readily seen that v E @In(G; (Da)l.;Bm-l). Using (8.3) we have :
.
zi
< , zw .
- -
(3
(z
a a.
One can take e mapping of the form : x, = 5, , ?-.,
(9)
...,
q).
SHMIJEI, AGMON: The Lp approach lo where co depends only on the mapping. Applying now Theorem 6.2 to the function ; in 2, we conclude that E; H j , L p for every r 1 and consequently that u E HjILp(go), @ being the image of Fr under the mapping. We also obtain by the same theorem the desired estimate. This establishes Theorem 8.1. From Theorem 8.1 one deduces easily the regularity up to the boundary of weak solutions of the Dirichlet problem :
(zr)
on
1Dau=O
aa,
z, such that I&,)d and z, z, in. the sense required. In the case of single integral problems, where
-
-
Tonelli (see, for instance [76]) mas ableto carry through this program for the case that only absolutely collti~~uous functions are admitted, the convergence is uuiform, and (esseatially) f ( B z ,p) is convex in p (ifj'(m ,z p ) lf,,(p) where f,(p)/ 1 p I o , it is seen from the proof of Theorem 2.4 below, that the functions in any miuiluizing sequence would be uniforlnly absolutely conti~~uons so that a subsequence wonld converge ui~iformlyto ; ~ uabsolutely conti~luousf u ~ l c t i ozo ~ ~which would thus lninitnize I(z)). Tonelli was also able t o carry through the eutire program for certain double integral problems usil~gfunctions absolutely continuous in his sense (ACT) and uniform convergence [77], [78]. However, iu general ha had to assume that the integraud f (x y z p q) satisfied a condition like
-
,
,
,, , ,
I f f satisfies this condition, Tonelli showed that the functions in any mii~imiziug sequence are equicontinuous, and uniformly bounded on interior domains a t least (see Lemma 4.1) and so a subsequence converges 1111iformly on such domains to a function still in his class. H e was also able to haildle the case mbere (0.3)
f ( x , y , ~ , P , q ) ~ 9 ~ ~ ( 1 " q ~ )k- if
f(",Y ,z,O,0)=0,
for instance by sllowing that any ~ninimizingsequence can be replaced by one in which each z,, is monotone in the .sense of Lebesgne (see (311 and [37], for instance) and hence equicontinuous on interior domains, etc. However, Tonelli mas not able to get & general theorem to cover the case where f satisfies (0.2) only with 1 < a < 2 Moreover, if one coiisiders problems involving v > 2 independent variables, one soon finds that one would have to require a to be > r in (0.2) in order to ensure that the functions in any ~ninimizingsequence would be equicontinuous on interior domains. To see this, one needs only to notice that the functions
.
a,re limits of ACT functions in which I, B(0,l)
'1
I
dx and I V .1
I* dx for k < v/(h + 1)
B(0,l)
respectively, w e unifor~nlybounded (see below for notation). In order to carry throngh the prograin, for tliese more general proble~ns, thon, the writer found it expedie~itto allow fuuctions which are still more ge~ienllthan Tonelli7s ACT functions. One obtaii~sthese more geueral fullctions by merely replaciug the requlre~rieut of v-dimensional continuity iu Tonelli's defiuition by snmniability, but retaining Tonelli7s requirements of tlbsolnte continuity along liues parallel to the axes, summable partial derivatives, etc. But then, two such fuuctions may differ on a set of iueas~lre zero in such a way that their partial derivatives also differ only on a set of measure zero. II, is clear that such functions should be identified and this in doue in forming the <spaces %A3 discussed in Chapter I. These inore general functions have bee11 defined in various mays and studied by various authors in various connections. Beppo Levi [32] was probably the first to use functions of this type in the special case that the fu~ictionand its first derivatives are in J2; ally function equivalent to such a function has been called strongly digerentiable by Friedrichs and these functions and those of correspouding type involvi~lghigher derivatives have been used extensively in the study of partial differential equations (see [2], PI, ill], [Is], POI, 1211, 1241, 1281, 1301, [421, [451, [46l, [471, [BOI, [671, [61], [66]), G. C. Evans also made use a t an early date [14], [16], [16] of essentially these same functious in connection with his work on poteutial theory. J. W. Calkin lieeded them in order to apply Hilbert space theory to the study of bouildary value problems for elliptic partial differential equations and collaborated with the autlior ill setting down a uulnber of useful theorems about these functions (see [4] and [40]). The fuuctions have been studied in more detail since the war by some of the writers inentioned above and by Aronszajn and Smitli who showed that any function in the space H,,,(see Professor Niremberg's lectures) can be represented as a Riesz potential of order m [I]. The writer is sure that many others have also discussed these functions and certainly does not claim that the bibliograplly is complete. I n Chapter I, the writer presents some of the known results concerning these more general functions. In Chapter 11, these are applied to obtain theorems concerning the lower-semicontinnity and existence of minima,
CHAET,ICS3. Moartlc~ JR. : Multiple integral
of multiple integrals of the form
where the function f is assumed to be continuous in ( x , z , p ) for all ( 8 ,z , p ) a'nd convex in p = (p:) for each ( x ,2 ) . I n Chapter 111, the most general type of function f ( x , z , p ) for which the integral I ( z 0)in (0.5) is lowersemicontinuous is discussed. I n Chapter IV, the writer discusses his rescllts concerning the differentiability of the solutions of minimum problems. I n Chapter V, the writer discusses the recent application by Eells and himself of a variational method in the theory of harmonic integrals, We consistently use the notations of (0.5). I1 p, is a vector, Ip, 1 denotes the square root 0.f the sum of the squares of the comphents. Our functions are all real-valued uilless otherwise noted. If x is a vector or tensor z, z a p , etc., will denote the partial clerivatives dz/dxu d 2 z/dxa a d , etc., or their corresponding gei~erltlizedderivatives. Repeated indices are summed unless otherwise noted. If G is a domain dG denotes its bo~indaryand G = G U dG. B ( x o R) denotes the solid sphere with center at xo and r:bdius R ; we eometimes a.bbreviate B ( 0 , R) to B R [a, b] denotes the closed cell aa l xu 5 ba. A11 ir~tegralare Lebesgue integrals. I t is someti~nesdesirable to consider the behavior of a function (or vector) z ( x ) with respect to a pa,rticular vari;ible xa ; when this is done, we write x = (la ,$A) and z (z)= z ( x u ,2;) where xi stauds the remaining variables ; sometimes (v - 1) dimensional integlals
,
,
,
,
,
8:)
dxi
a,:
appear in which case they have their obvious significance. We say that a (vector) fnnctioi~z (8) satisfies a uniform Lipsohitz condition on a set 8 if snd only if there is a constant M such that
I z (xi)- z (x2)I IM I xi - x2 I for x,
and x2 on 8 ;
x is said to satisfy a uniform Holder condition on S with exponent p , 0 < p < 1, if and only if there is an M such that
I z ( x , ) - z ( ~ , ) I < X ~ 1 x -x,Ir ,
for xi and
l,
on 8 .
proble~nsin the calculus etc.
A (vector) function z is of class Cn on a domain 12 if and only if z and its partial derivatives of order I. s are coutinuons on G ; x is said to be of class 04+r or Cn on CS if and only if z is of class 0" on 0 and it and all of its partial derivatives of order I. s satisfy uniform Hiilder conditions with expoueut p , 0 < p < 1, on 0 ; the second notation 0; is used when p = 1 (see Chapter V).
Function of class qA, %i ,93;(11 1) and functions which are ACT. We begin with the defllritio~isof these classes: DEFINITION: A f~lnctioli z (m) (x = (xi, xv)) is of class oe a do~saijc U if and only if z is of class on G and there are f ~ l ~ ~ c t i op n* s, u =I , Y of class &A on G with the followillg property ; if R ie any cell with closure iu G , there is a seqtle~rceZ,~Rof fouctions of class 0' on H U dR such that s,,- 2 and zn,,-pa strongly in 2' on R. DEFINITION:A fiinctioll 21 is of Class %?i OU C f i f alld only if (i) a is of class &A 011 G ; (ii) if [ a , b] is m y closed cell ia (f tlleli z is AO (tibsolutely continaous) it1 xa on [au, b y for i~lniosta11 xi on [ a : , b:] a = 1 v; (iii) the partial derivatives a,, which exist' almost every-where and are measurable on accoant of (ii), are of cltiss &A on G . DEFIN~TION: A fuuction z i s class 9;011 B if and only if z is of class on G and is continuous there. D E F ~ N I T I ~A N : functiotl z is abaolt4tely co~ctinuous in the sellse oj' Toxelli (ACT) 011 G if and ouly if z is of class 93; ilncl is oontinuous on G . DEFINITION: Suppose z is of class &, on 0. We define its 1~ average function on the set Gh by
...,
..., ,
,
,
,
, ...,
(v
ith
zh (x)= ( 2 1 ~ ) -z~( E ) 115
,
X-h
+
Gk being the set of a11 x in G such . tlrt~tthe cell [s - lb, x h l c G. LEXMA1.1 : I s z ,is of class ZAole a d o l ~ t a C G trnd Zh i s its 1b.avevage as h 0 09, each closed cell [ a , b] fi~lcction defined on Bh thetb zh z i a 4 G awd zh i s contittuous O I L ah. Proof: That ah is continaoas follows from tlre tlbsolute continuity of the Lebesgue integral. Next, it is well known that zh (x) x (m) as 1b 0 for almost all x. Finally, choose IL, > 0 so tlltit [a - ho b k,] c G , keep 0 < k < lc, , and let y (g) be ii function 0 as g 0 such that 11 z lie 5 y [m (e)] for e c [a - lb, b Lo] where
,
-
-.
-
, + ,
-
-
, +
-
problems ithe calculus eto. Then the le~~lrna follows, since since
where e(5) is the set obtained by translating e along the vector 6. TEEOREN1.1 : I f z is of class q Aon G , the functions pa are cvnipael?y detevrbined up to null functions. I f zh is the k average of a attd pah is th(rt of pa then zh is of class G' on C f h and
,
, +
,
,
Proof: Let [ a , b] c U choose W, so [a- h, b JL,~c G imd keep 0 < h < h,. Approximate to z and pa by z, and x,,,, ia 2 2 on [a - h, b W,]. Then for e ~ c l lh , me see t l ~ n t aflh;,= (z,,,)~ and me lnay obtain (1.4) by letting a - oo on [ a , b]. The first stictement is now obvioas. DEFINITION : If z is of class %IA on a domain G , me define its generalizet derivative D, z ( x ) as the Lebesgue derivative a t x of the set fuuction
, +
/pa ( x ) h.
THPOREX 1.2 : I f z Xi8 of C ~ Q S S % oni Cf, zh is its h-average ficnctiox, and pah is that of its pavtial deritative &/axa, then zh is of class Or and (1.2) holds. Moreover z is ?f class %Aand its corvesposding partial awd generalized derivatiyes coincide a111tost evevywhere. Pvoof: Let [ a , b] c G , choose W, so [a - ho b k,] c U and keep 0 < h < h,. If x: is not in a set of measure 0 OII [a: - I&,,b: h] then azlaxa = p a ia summable in ma over [aa- hq ba W.,] alrd
,+
, +
,
+ ,
By integrating (1.3), we see that i t holds for all x: on [a;, b:] and all a ; , x; on [ a a ,ba] if z and pa are replaced by their h-averages. hen (1.2) aud ,the laat statement follow.
C H A R L ~ SB. Moarte~J a . : Multiple integ~al
THEOREM1.3: (a) I f zi atcd zz trre eqccivalent and one i s o f class on 6 , then both are and their ge~re~alizeddersiuatives coincide. (b) I f z, and s, tare o f class 9& O M a doa~aitt G and z,,, (x) = (3) a111lost everywhere o n 6 , t h e i ~ z, and z, dbfer by a consttrnt and a null .function. These are easily proved using the haverage filuctions. THEOREM1.4 : (?) A n y function z o f class on G i s equivalent to a function a, o f class 93i on Q b) z i s A C T ota G if and only if z i s of class 93; there. Proof: To prove (a), let R = [ a , b] be any ratiolinl cell in G and approxi~nateto z there by functions z, of class 0' on [a, bl. A subsequence, still called z,, converges to z almost everywhere and is such that
.
(ma n-w
,$3- z,, (aa, 4)IQxa = 0
E
, ,
for it11 x; uot in a set ZRa of ( V - 1)-dimensioual measure zero, u = 1 ... v. Prom (1.4), me see that the x,, (xu ,xL) are equicontinuos in ma and converge uuiformly on [ a a , b*] to a function XOR (xu .c;) which is A C in za if 3: is not in ZRa, u = 1, Y. Obviously zoR= x almost everywhere on R. Since the uuiou of the ZRafor u fixed ancl K running over all r;ttional dells is still of measure zero; we see that the $ 0 join ~ up to form a function 2, of cPass 93 ou G. To prove ( b ) , we note first that if z is ACT on 6 , it is of class 93; on G. Conversely, if x is of class qy,we may repeat the first part of the proof taking z, as the lh, -:tversge of z and conclude that we may take z o ~ always = z since then z,, couverges u~~ifornily tJo z on R. T l ~ efollowing theorems are easily proved by approximations : THEOREN 1.5 : l'he space 93, o f eqllic~tleaceclasses o f functiotis of class % i s (G Banack space we define the ~aovqnby
... ,
I f rZ = 2 ,
,
i s a real Hilbert space (f we de$tte
problems in the calcultts stc. THEOREM1.6 : I f t2 E aild i s qf c l a ~ sC' alrd sfitisfles n tc!~ifOrv)t Lipschitz condition on the bouirded donrain G , then i~u 6 % ow Q and the getteradized derivatives (hu),, all exist at nny point x, where all the zc,,(x,) exist. onto ff I)EFINITION : A trtlusformation T :x = x(y) from a domain which is of olass a ' is said to be regular if and olily if T is 1- 1 and T and its inverse are of class C' and satisfy uniform Lipschitz condition (1 x (Y,) - 8 ( ~ 8 I)S31 I Y, - Ye I, etc.1. THEOREN1.7 : I f u i s of class (93;)olt the botcnded demain ff, x=x(y) i s a regular trnnsfort~aatiolzof olass C' from the bounded dontaii~2 ofit0 Cf alzd (y) = u [x (y)], then i s of class (%:) on Cf Moreover, if ~ , = x ( y ~ ) and all the gelieralized derivatives u,, (x,) exist, then all the generalized derivatives G,, (yo) exist atzd
.
-.
P r o o f : That % is of class %I(%;') and that we may choose the right sides of (1.5) as the ct derivative functiolrs B of the definition is easily proved by npproximatilrg u on interior domains by fuuctions of class U'. S i ~ ~ cregular e families of sets correspond under regular transformations, the last statement follows easily. REMARKS : I t is prqved in [40] arltl [47], for instance, that if zc is of clttss %A on a, it is equivalent to a fu~iction (namely the Lebcsgue de-
FP
./
rivetive of udx) which is of class
x
% and i is such that any trausfor~rias
in
Theorem 1.7 retsins this property. But the last statement of Theorems 1.7 does not hold for the partial derivatives since this wonld imply that z had a total differentiill almost everywhere central-y to an example of Sake [55]. I t is clear how to define the gerleralized derivative in a given direction and that (Theorem 1.7) if 2111 the u,,(x,) exist, then a all the geiaevalixed directional derivatives exist at x, a ~ are ~ d given by tlreir usual formulas there. It is now easy to prove Rademacher's famous theorem [52] that a Lipschitz function has a total differential almost everywhere : For lrsing the result just nle~rtio~~ed together wit11 Theorem 1.2 we see that if z is Lipschitz and x, is not in a set of measure zero, then the partial and generalized derivatives all exist at xo and the ordinary directional derivatives in a denumerable everywhere dense set of directions (independelrt of 3,) all exist and are given by their usual formulas; at ally such poilrt z is seen to have a totd differential. Thus in Theorem 1.6, h may be Lipschitz illid in Theorem 1.7, the transformation and its inverse 1na.y be Lipschitz; in this case (1.6) holds whenever a11 the geaavalized derivatives involved exist.
CHAKLESB. MORRDYJR.: Nultiple integral
THEOREM1.8 : The ~itostgeneral lkear fictcctioaal on the space the form
% is oj
4
where the A, (a 2 0 )E with 1-1 f p-1 = 1 i f 1> 1 or are bounded and ~~ceasurable ole G if 1 = 1. Proof: Let An be the space of all vectors y = ( y o , ,y,) wit,h components in and
...
,...,
From Theorem 1.5 it follows tl~tbtthe subspace of all vectors (z,z , ~ z,,) for mlrich z ou f f is a closed linear manifold M i a Bn . Hence if P (2,zgl z,,,) =f ( z ) the11 P can be extended to the whole space B to have same norm aef. Then P is given by (1.6). From Theorem 1.8 we immediately obtain: THEOREM1.9 : (rc) A Necessary a ~ szcflcient ~ d co~cditiojcthat z,, converges on G. wecr,kly to z (z, 7 8 ) ia % on G is that z, 7 a a~rdthe z,,,, 7 z,, C (b) I f zl,7 . 2 in %Aos a, theit z, 7 x in %AO D nny subdoweni~. or f f (bounded), x = x ( y )is a regttlar trassforma(c) I f z, 7 z i n w tion of class 0' from owto ff, ; ,( y )= X , [x (y)] and z ( y )= z [x (y)],then F, 7 Fin on a. (d) I f a,,7 z i n qnon f f (bounded) and h is Lipschitz on f f , thew hz, 7 ha igc %Aon ff. DEFINITION:A fu~ictionz is of class 5Yjio on G (bounded) if and only if it is of class C)& there and there eviste a seqnence { t i n ] , each of class C' and vanishing on and near tlre bolllldary dG such tllat 2,- z (strong convergence) in % on ($. The subspace %A,lof %Ais defined correspondingly. If z i1.11d2"E %Aon G, we say that11 z = z* on dG itc the %a sense if and only if z - z* E CX3no on G* The follawiug is imn~ediate: is a closed liltear ntalcifold ,is %A; THEOREM1.10 : The sttbspace qAo ifz,, 7 z be %Aoic G a d each z, E%,,then z E %lo. I f z E 9 1 0 and z, (8)= z(x) for o on G and z, (x) = 0 otherwise, then z, E %'A~ on auy I) 3 G and z,,,(x)= 0 for alntost all x not i ~ tG. THEOREM1 .ll (PoincarB7s inequality) : Suppose z E %AO on GC B(xO R). Then
,...,
,
%
,
pvoblen~sin the cnlczrlus etc. R o o f : I t is s~ifficicient to prove this far x of cl:bss C' and va~iishing on dB (so,R) with G = B (so,22). Taking spherical coordinatei3 (v,p) with r = I x-xol and p E 2 = d B ( 0 , I ) , me obhin
where
21 (1.
,p) = 2
(3). Tl~iis
from whiclr the result follows. on 6, A c Q z' E %A 0 t h /1 trnd coinciTHEOREM1.12 : Alppose z E des with z on dA ill, the % se~ue.Then the ftcvtction Z such that Z(X)= zY(:z) on A altd Z ( x )= 2; ( x ) on Q - A i~ qf cltrss o s a irnd z,, ( x )= x:(x) nlnzost euaryzohere O N A alcd Z,, ( x )= z,, (3) trlntost ccverywheve on G - A Proqf: For define 2, ( x )= a" (x)- x ( x ) on A i ~ l d0 elsewhere. Theu Z ( x )= z (x) Z, (a) on G a11(1the res~iltfollows from Theorem 1.10. LEMNA1.2 : Suppose z E 93~on tlrt: cell [a - h, , b h,]. Then
,
.
+
+
6-l-h
zh(x)-~(x)~*dx~Ci(v,~).~~L.~ll~z(y)lAdy, a-h
where Ci depetids olaly ola the argztsaents indicated. Proof: Since we may approxin~ateto z stroi~glyin %Aon [a- h, b h] by f~inctiollsof class C' on that closed cell, it is siifficient to prove the lemma for sucl~functions. Then if 3 E [a, b] and 1p ( 5h , we see that x and
+
CHARLESB. MOKEEY JR.: Multiple integral
104
a + [ are in [ a - h , h + b ]
so that
Then
+ [) - z (x)]df IA dx
(z)- z ( 8 ) I A dx =
[z (x -h
from which the result follows. TFIEOREN1.13 : If z, 7 zO i n qoA O N the bou~tdeddoniail~G, thea 2,-z, iu JAon G, ,I> 1. I f (z,,)is IG seque~toe in %oA tuitlb (1 zn 11 unijor11tly bot~nded, a stcbseq~beltoeoonuergec strongly ill 21to some futtotion z. Proof: The first statement follows from the second. For, let (z,) be any snbsequence of (z,,]. A subseqrrence { z g ) converges strongly in 2 2 to some function z which must be (equivalent to) 2,. Hence the whole sequence z,, zo ill gA. To prove the second sttttement, suppose (f c [ a , b] and extend each z, 1 with uniformly to be 0 outside 0 ; tl~ell each z,,E C M o ~ 011 [a - 1 a 1 bounded 91ilorm. For each h with 0 < h < 1, me see that the Znh are uniformly bounded and equicol~ti~~uous on [ a , b]. So there is a subsequence, called {z,), such that zph converges uliiformly to some function zh for each h of a sequence 0. Froln lemma 1.2, it is easy to see first that the limiting q form a Oauchy sequeuce in @. A having some limit z and then that z, 2 strongly in JL. I n order to treat variational problems with fixed boundary values, one can, of course, practically always reduce the problem to one where the given boundary values are zero. Althoiigh one call formul:~te theorems about variational problems having variable boundary values on the boundary of, an arbitrary bounded dolnltin (see Chapter 11), suc,h problems become more
-
, +
-
-
problem in the calculus etc. meaningful if we restrict ourselves to domilins U which aro bounded and of class Cr where bouudary values can be defiued in a more definite way as we now do. DEFINITION:A bounded domain G is of class 0' if and only if each point xo of the boundary dQ is interior to a neighborhood N ( x O )ou G a O which is the image, under a regular transformation x = x ( y ) of class C', of the half-cube Q+ : I ma ( < 1 for u < v and 0 5 xv < 1 , where x ( 0 )= xo and dff fI N ( x o ) is the image of the part of Q+ where xv = 0. Such a neighborhood N(s,,) is called a boundary seighborkood. DEFINITION : Suppose G is a domain. A finite sequence (h, h N )of fuoctions is said to be a partition of ullity of class Or on G U dG if i~nd only if each hi is of class C' on G'UdG, O l h i ( x ) l l on O U d a for each i,and
u
,... ,
N
2 h i ( x ) f i for x on O U d O .
i=l
Tlie support of 1bi is the closure of the set of all x on O U dG for which hi (x) 0 LEMMA1.3 : If G is botrltdrd dol~~tti~b of clnss Cr, there is n pnttitio~of tiftity (li, liN) of cltrss Cr on U U dG sucW flint the suppovt of each hi is either interior to a call ilc G or is inta~iov to a botcfbdary ~teigliborlboodof G U dG. Proof. With each interior point P of Q we defirle Rp a8 the largest hypercube Ixa-$;I 0
for all ? on B and all 17. DEFINITION: A linear function a, 6, b which satisfies (2.1) for some 2 is said to be supporting to p, a t ?. LEMMA2.4 : Jlzppose p, is oonves for all 6 and satis$es
+
, ,.. ,
Then q~ takes 0th its ntiaimzcm. Also, if ai ap are any numbers, there is a unique b such that a, EP b is supporting to p, for some E . I f y, is convex and satisjies (2.2), if jf (6) 2 p, (6) for each 5 , and if ap 5p c is snpporting to y , then c x b . LEMMA2.5 : Buppose that p, and p, are everywhere convex and satisfy (2.2) and suppose that p,, (t) p, (6) for each t #upposs a, , a, are any nun8bers and b, and b are chosen so that up 5 P b, and a, 6p b are s ~ p portitag to rp,, and p,, respeotively. Then b, b Likewise, i f a, -ap for each p and b, and b are chosen so that a, 6 P + b, and ap EP b are all supporting to f, the* b, b . I n order to consider variational problems on arbitrary bounded domains, it is convenient to introduce the following type of weaker than weak convergence in 33, on such a domain. DEFINITION:We say that Zn z0 in %, on the bounded domain G if aud only if z, and zo all E %, on B , zn 7 z, in %, on each cell interior to C: and each z , , 7 zo,, in 2,on the whole of G THEOREM2.1 : If Q is bouaded and of class C' or i f all the z, E %,, on 2, in 33, on Q , then z, zo in qi on 6. Q and if z, Proof: The second case can be reduced to the first by extending each z, to be zero outside G and choosing a domain r of class 0' such that T DG . Thus we suppose CS of class a'. If we use the notation in the proof of Theorem 1.14, we see that (1.7) holde uniformly for the wn, so that an arguu~elit siluilar to tilose in the proofs of Theorems 1.14 and 1.15 and 1.13 shows that w,,i converge strongly in k?, on Q or Qf to something for each i . Thus z,, converges strongly in Ji on Q to something which must be z, REMARK:If Q is not of class 0' and the 2, are not all in qio on 8 , the11 an example in [41] shows that z, + 2, in 9, on Q w i t h o ~ ~ t
+
+
-
,...
.
-
.
+
+
+
-
.
T)
.
CHARI.ES B. MORRICYJR. : Multiple integrat
the
q, norms of the zn being uniformly bounded. If for some 1>1,
11 -1 F h , dz
G
[i;:,.Py
( C bounded)
0
are uniformly bounded, the11 a subsequence { p ) of (s)exists such that the z,,, 7 something iu 2, on the whole of C . THEOREM2.2 : fJ2cppo~ethat f ( p ) is dejlzed of all p = ( p i )(i = 1 , N a =1 v) and f is convex. I f z, + zo oil C and
...,
, ... ,
,
thelz I ( x , G) and I ( z , , Q) are each j ~ i eor
,
+ oo and ,
I (z, C)5 lim'inf I (8% 0). n-to
Pvoof: Since f is convex, there are constants a; such that
for all . p Hence
with
R similar inequality for I ( z , ) . Thus the first statement follows. If D C C we see as above that
,
by virtue of the uniform sbsolute continuity of the set functions
,
,
(z)d z e
.
Clearly also I (z D)- I (z C ) as D runs through an expanding sequence of domains exhausting C . Thus i t is sufficient to prove the lower semicontinuity for G a hypercube of side h , say. To do this, we define a sequence of summable fulictions y q ( x ) as follows: k'or each p divide C into 2'9 hypercubes of aide h 2-9. On each
problems i# the calculus stc. of these hypercubes R , define
where pia is t h average ~ of zfa over R and the a; (pi -pia) is supporting to f a t p~ larly from 2 , . Then i t follows that
+
f (pH)
id;
.
( R , q) are chosen so that W e define the pl,, simi-
(almost everywhere). On the other hand, suppose all the genertdized derivatives exist a t some xo which is not OIL d R for any hypercube R as above for any q . Let R denote the llypercube containing xo Then as q oo pia nfa (8,) so that yq (8,) f [Vz (a,)] since the a; remain bounded (Lemma 2.5). Hence
IVq
~ ( z 8) , =~ i m 9. 6 '
(2.3)
-
.
-
pa-
h.
,
Moreover, for each fixed q p , --pR ~ f r o ~ nthe weak convergence so dx = 2 f (pR)111 (R) = liln 2 f (pllR)111 (R) = R
G
=liln
11-m
I
R
yfsq(x)d x ~ l i l n i n f I ( z , , , O ) .
n-w
n-w
G
The result follows from (2.3) and (2.4). LEMMA2.6 : Suppose j'( a , z p ) is dejiaed aid, satisfies n unifovolal Liyschitz coitdition with consttrv~tK for all ( a , z p ) szbppose f (x z p) i s convex iia p for each (a, z) aed suppose f (x x p ) 2 f,(p) f b r n l l (m x p ) , where f, (p) is convex. Then, if x, 7 zo in on CS
,
, ,
Ti
,
, ,
, ,
, ,
,
I(#,, C) 1 , let E, be tlre set of x in G where r - 1 5 1 V 2 ( x )I< r and V z ( x ) exists and let
where Z is the set of measure 0 where V z (a) does uot exist. Clearly Lo = G and if r~ 1 a1111 $ 8 O - &,, theu I V #(%)I 0. Then (3.2) holds for all ( x x ,p) on & Proof: For, let 5 be any Lipschitz functioli vanishing vanishing on and near dB. hen 2, 15 is sufficiently near so for a11 sufficiently mall
v
I
+
,
+
v
I
Csaar.vs B. MORRICYJR. : Nultiple integral
1 So if ~ ( 1=) l ( z o
+ A[), we must have
By selecti~~g any point xo in G and proceeding as in the proof of Lemma 3.2 alrd the11 dividing by Rv/v (v - I)], but letting R and h both 0 so that k : B 0 , we obtain (3.2) at [x,,, 2 (xo), p (xO)], Using the result of Le~nlr~tb3.2 and the method of proof of Theorem 3.3, we conclnde that i f f ( p ) is quasi-convex and of class Cn, then (3.2) holds wit11 s and z omitted. This ~.esnltand the analogy wit11 convex functions suggest the followi~~g theorem whichwe IIOW prove. THEOREM3.4 : I f f ( p ) is quasi coltvex, the14 f ( p i 1, lj) is convex i n 1 for each p and 8 and convex i n 6 for enoh p and 1. Proof: I f f is quasi-convex, it is easy to see that its twiceiterated haverage f11nctio11jhh ie a180 quasi-convex and is of class C" as well. Then any linerir function furnishes an absoliite minimum to Ihh ( 2 , a) among all Lipschitz functions with the same boundary values. Accordingly, by Tl~eorem 3.3 we see that fhh satisfies (3.2). But then fhh has the convexity properties stated in the theorem, Since .rhh collverges uniformly to f on any bouuded part of space, the theorem follows. DEFINITION: A function f ( p ) which satisfies the conditions in Theorem 3.4 is said to be weakly qnasi-convex. REMARK : The principal problem, so far unsolved, is whether or not every weakly ,quasi-convex function is quasi-convex. THEOREM3.5 : If f ( p ) is weakly quasi-convex , it satisfies a unifornb Lipschitz condition on a bounded pcwt of space. I f p is given, there are coltstants A; such that
-
-
[r,
+
f ( p h + l a € j ) >-f ( p ! ) + A;IZa[j for all 1 , t . I f f is also of class O', then Aa==f,i(p). I f f is also of class CN then (3.2) 3 a holds. I f f is continuozcs and if, for eachp, constants A4 exist such that (3.8) holds, then f is weakly quasi-convex. Proof: I f f is weakly qua,si-convex, it is convex in eachpi separately. Hence, if If ( p ) l i ; M on some hypercube, any difference quotient of the form :
I [ f (PA)--~(P!~)]/(P,',-pja) I 5 2Mld ,PA < P!: where d is the smaller of b! -p,iQ and p i
- ad.
problems in the cnlc~ilusetc. Next, Jhh is still weakly quasi-convex and of class C" so that (3.2) llolds. Then, frolo the co~ivexityin 5 for each I, for instance, (3.8) holds \vit,h A& = fhhp"p). Si~lcef satisfies a ul~iform Lipschitz condition near p, we see that the Aihj are u~lifor~nlybounded as h 0 so a seqile~iceof 8 - 0 ~2111 be cl~osellso that a11 the A;thj tend to limits. Olearly (3.8) holds in the limit. Since tile unit vector in the pi direction is of form 1, t j , we s e e that A; =f p j if j' is of class C'. The last statement follows from ,theorems 011 convex functions. We now define a sufficient condition for f to be (strongly) quasi-convex. THEOREM3.6 : A stcflcieiit cosditiou l o r f to be qucrsi-convez.is tlrut for eitch p tibere exist alternating forms
-
Q
(in which the coefficients ere 0 unless all the a , ...a, are distinct and all the j , ...jP are distinct and an intercllauge of two a's or two j's changes the siga) szccl~ that Jbv ull n we lurve
Proof: For snppose p is any constarlt tensor, G is any bon~~ded domain, a ~ i d[ is ally Lipschitz vector whicll vanishes on dG. By extending 5 = 0 outside G a ~ l dapproxi~ni~ting to it on a larger domain D wit11 srnootll bouiidiwy with fui~ctiousof class C" which vanis11 on a ~ ~near d dD and using Stokes7 theorem we see that the integral of the sum on the right in (3.9) is zero. We now exhibit two iuteresti~~g cases where the weak quasiconvexity o f f ilnplies its quasi-convexity. THEOREM3.7 : Ij' f ( p ) is weakly quasi-oonvex and
tlten f is quasi-convex (1791, [45]). Proof: For, if 5 is Lipschitz and vanishes be assunled smootlr), then
011
dG (which mzly as well
CFIARI.ES B. MORREY Ju. : Multiple intsgvai If me introdace Fourier trltnsforlus (see [79])
we see that
since the i~ltegrandis 2 0 for each y. THEOREM3.8 : If N = v 1 and
+
where P i s continuous and
...,
Then f i s quasi-couvex ht p if and only if P i s convex iw (4, Xy+l). We omit the proof which is fouud in [44]; P is there ~eqhiiedto be lio~nogeneousof the tirst degree in X but this is not necessary in the proof.
problems
(11
the enleulzts ete.
The differentiability of the solutions of certain variational problems with v = 2. 111 this chapter we disciiss the differentiability of the solutions of certain proble~nswhose existence wits proved in 5 2. To save time, me shall not discuss the coutinuity an the boundary bnt shall consider only the differentiability on the interior. This work ww,s first presented in [42], chapters 4,6, and 7 and was the culmination of s series of papers on this subject by Licl~tensteill[34], [35], Hopf (271, imd the writer [39]. Some of these results have recently been generalized by De Giorgi [lo] and Nash [49]. Sigalov [Gl] t~nnouncedresults siinilwr to those presented here. We begin with the following lemma which has a proper'generalization for all values of Y (see [42] and [47]): LENMA4.1 : S'uppose a vector z (x) E 9, O N n domain O and suppose that
for O < r < n ,
(4.2)
1 s (2,)- s (xi) I 5 Ci ( I ) . L . ([xi- ~ ~ l / afor) ~ 0 1 xi - xg 1 5 a ,
where Ql(A) = %1-"--1/2
1-1
,
for every pail. of points (3, xz) in O such that every point on the segmeirt joining them is at a distalbee > a from d B . I'roof: We note first that if E is on the segment and s 5 a ,
j
1 V z (y) 1 dy 5 dl2La-bl+',
B(P,8)
CHARLNSB. MORREY J R . : Multiple ilztegrat
128
using the Schrarz inequality. Next we write
+
I 2 ($2) - 2 (xi)I 5 I (4- 2 (x,)I I ($1
- 2 ($2) I
:
6,
,
+
and then average with respect to x over B r/2) = (xi x2)/2,. I f given t 0 < t < 1, we set y = xk t (x xk), then y ranges over for a B [(I- t)xk t i , rt/2]. Then
+
,
+
from which the result followa. NOTATION: If z€CM2 on G , we define D ( z , G ) =
called the Divichlet integral. LEMMA4.2 : #uppose n E q2on B (so, a) and suppose
where
.
onnuerges, for every .function 2, = z on dB (xo,r ) Thehell
and the right side tends to zero with r.
IVz12dx; this is
problems in the calculus etc.
.
,
Proof: Let p ( r )= D [ Z ,B ( x , r)] Then p is absolutely continuous. For almost all r z (r , 0 ) is AC in 8 with I z, ( r ,'0) 1 in For such r , define
,
4.
Using Fourier aeries, one easily sees that
/" 1
2 (r 9
0 ) - ;(I.)
l2
5
/" 1
2,
,
( r 0) 18 d0 5 r pf ( r )
By computing D2 [Z,, B (m,, r)J and using (4.5) we see that
from which (4.4) followa easily. In order to see that the right side of (4.4) tends to zero with r , we note that
,
THEOREN4.1 : Suppose f ( 8 ,z ,p) is continuozls for all (3: z ,p) alta is convex in p for each ( 8 ,z) , asd suppose there are oonsta?zts 118, M and 1c such that
.
,
,
for all p Suppose I ( # , U) is finite, Q is a bounded domain, and z, miaimizes I ( z , U) among all z in 93%coinciding with x, on 86. Then z, satisjies (4.1) and (4.2) on Q with
I s s / 2 M and L 2 = D [ z 0 B ( r x : , , a ) ] + 2 k n d / M . Tlbzcs 2, satisfies a unifornt Holder co~rditionon each colitpact szcbset of 6 . Proof: Suppose B (a, r) c Q and let Z, be ally function in 33, on B (so,7') and coiuciding with ' z , on d B (x, v) Then, from (4.7)
,
,
, .
,
,
I,ID[z, B,.] - kn r2 5 I (zo Bv) 5 I ( z r B,) 5 M D (2, Br)
The result follows from Lemma 4.2.
+
v2
For the remainder of this section, we shill1 asstune that f ( x , z , p ) sa. tisfies the following condition in addition to (4.7): GENERALASSUMPTIONS: W e assu?n,e that B i s a bounded do$)iain,f satisjies the conditions of Theoreni 4.1, and (i) f i s of class 0" for all ( x , z , p ) (ii) there are functions m, (R) M, (R), and n12 (R) with 0 < vr, (R) < M,( R )for all R 0 such that -
,
at, (R) I n l2
0 0 r 5 n and Po on M, there is an adwzissibile coordinate system mapping B ( 0 , e), for some e > 0 , onto a ~zeiglbborhood U of P o , and a constant 1 such that
,
,
J
2 D ( w ) ( 1 - E) (i)a B(o,e)
CO?~),,
,
dx - 1 (w w )
332
for any r-form E whose support is in U. Proof: We begin by choosing a fixed coordinate system mapping some BR = B (0, R) onto a neighborhood UR of Po,carryng the origin iuto Po, and satisfyng go ( 0 ) = d i j . From our formulas for dw and 6 0 , we see that (5.24)
+
J
D (o)= [a(W)aBoci),tz o(j , , ~ 2b@)( j l a o ( i ) , aO(j)
+ ~ (( f0l wci,w(
j)]
dx
Be
where the a's are combination of the gu only and so are Lipschitz and the b's and c's are combinations of the gij and their first derivatives and so are bounded aud measurable at least. Since the a's are Lipschitz and since
me see that we may choose
Q
so small that
The result follows from (5.22). The following important theorem corresponds to Garding's Inequality for differential equations : THEOREN5.4 : For each r = 0 , n and coordinate covering C2e of M , there exist constants K y > 0 and Lg such that 0
,...
for ever w E %. Proof: Prom Theorem 5.2 it is sufficient to prove this for some particular q.Let ?e = ( U i UQ)be an opeq covering of M by coordinate 1 patches such that ench ,c€ M is in some Uk satisfying (5.23) with 8 = 2
, ... ,
'
CHAR^.^^ B. MORREY
, ... ,
JR. : Multiple integral
say. Let Cf, U Q be the domi~in in E" such that Uk= Qk (Gk) for a11 k . There exists a finite sequence @, ,a,of Lipscbitz functions on ]I[, each of whioh has support interior to some Uq, nnd such that
, ... ,
for all x E M. Now if (5.25) aere false for the Q just described, there would exist a sequence (up)of $'-for~usill %; 811ch that D(up)aud (up, cop)mere uniforlnly bounded but 11 0.1, l r r - w Then, for some s p and some subsequence, still called u p ,me mould have
, ,
.
mhere @, has support in Uqt since
and
But it is easy to see that D (@, w,) and (as u p ,QS op)are uniformly bounded. From our choice of neighborhoods we have reached a contradiction with the fact that
We can now present the variational method. We begin with the following lemma : LEMMA6.2 : Let % be any closed linear manifold it, the space 2;of y;fovnts on N (of some one kind). Then eithev there is no f o m o of C)j7- which is in % or there is a form o, in %%; with ( m , , o,)= 1 which minimines D (o)among all such fornts. A ~ o f If : 972 contains no form in %;, there is nothing to prove. 0. thenvise let be a minimizing sequence, i. e., one such that ( a kmk) , =1 slid okE 9 ' 2Il9; for each ?c = 1, 2 and such that .D(ak)approaches its iufimulu for all u € % n %:. Prom Theore111 5.4 it folloms that the
,... ,
problenb in the calculus etc. ((wk,wk))ze are uniformly bounded. Accordingly, a snbseqnonce, still called (wk), exists which converges weakly in %' to some form w,. But from Theorem 5.3 wk tellds strongly in Pi to w, and 7) ( w ) is loaer-semicontin u o ~ ~with s respect to weak convergence in 93;. The proof of the lemma is now complete. DEFINITION:A harntonic yield w on d l is a form ill CM, on J4 for whicl~ dw = 6 0 = 0 almost everywl~ere..We mill let W denote the linear manifold of harmonic flelds on !I. of degree r . (Strinctly speaking we have %[ and for even and odd fonns, respectively). THEOREN5.5 : For each 1. = 0 n ( = din M) the littear lnaizifold W i s finite dimensional. Prooj. The 932 forms are dense ili E l , since the Lipschitz forms are. Let MI = 2;.There is a form wl in MI ll 93; which minimizes D (w) among all such forms with (ci, 6 ) = 1 Let $1, be the closed linear manifold in 2; orthogonal to w l , a l ~ dlet wz be the correspo~tdillg miliimizing form in M , . By continning this process, we may determine successive minimizing forms o, w,, w, , each satisfying (wk wk) = 1 and beillg orthogonal to $111the preceding ones. Now if ll (mi) > 0 , there are no harmonic fieltls 0 since D (w,) 5 ID (w,) 5 On tue other hand, suppose D (wk)= 0 for all values of K . Then by Theorem 5.4, ((wk wk))?( is uniformly boullded in k whence a subsequence [ a p )converges weakly ill 93; and hence strongly in 2; to some form wo in %;. This is impossible since the wk forrn an o~thonormal system in 2;. THEOREX 5.6: For each coordinate coveiing % of M there i s a oo?istntzt A, stwlb that
, ... ,
.
,
,
,
... ,
....
+
,
,
for any w in %; wich i s orthogolaal to %' . ProQf. For, let wo be that form in %; (there is one si~lceeach harmonic field is in q2) which minimizes D ( o ) among all o i l l 9; wit11 (w ,w)= 1 and o orthogolial to W . Then clearly D(w,) > 0 and by hol~iogeneity
for all o in %; and orthogonal to
from which (5.27) follows.
qr.By Theorem 5.4 we see that
CHARLESB. Mottss~Jlt.
:
Multiple kbteyral
TnEoREM 5.7: Suppose coo is any forwt in L?; crfrd ortlrogottal to 9[r Therb there is a uttigue form Qo in 9: and ort1bogollal to qr such thut
.
(a Qo, a 0 -t @Q0,dr) = ( 0 0 , 5) for every C in 93;. Moreover, the tratt.pformation front wo to Qo is a bou~tded linear tmnsformation from &; into Proof: From Tt~eorem5.5, we see that
.
,
since (w wo) is a bouilded linear fuactional on %; here 11 w =((w,s))qE. Hence I ( w ) is bounded below and is lower~semicontii~uo~~s with respect to weak convergence in %; if w is orthogonal (22-ser~se) to C3e". Accordingly there is id minimizing form Qo. If 5 is any form in orthogoultl to %I' we then see that
,
which shows that (5.28) hold's for all such 5 and Qo is nnique. But then (5.28) holils all 5 in since any such 5 is uniquely representttble in the form 9 = Il+ 5, where dH = 8H = 0 and to is in 9; a11d orthogorial to q v . Finally, if we set [=no in (5.28) aud use Theorem 5.7, we see that
from which the last statement follows. DEFINITION : The form Qo of Theorem 5.7 is called the potential of coo . We observe that if all forms in (5.28) and the mallifold M were sufficiently smooth, the equation (5.28), together with equation (5.18) would i ~ a ply that
I n any coordinate system, (5.30) reduces to a system of second order eqoations iu the components of the forlus; if r 2 1 , these equations involve the second derivatives of the gij as well as those of the components of Do. However, all the results stated so far hold for mauifolds of class 0;in which case the requisite second derivr~tivesof the gq certainly do not exist.
problem in
tH8
caleultrs etc.
,
DEFINITION: We say that o i s of olass 0 5 1< 1112 if for each coordinate system 0 with domain B R , there is a constant 1, = L (8,w ) such that
The olass 3321is defined si~nilarly. The importaucd of the spaces %Izn ariees from the fact that if w E ()32n with I = p - 1 n / 2 , O < p < 1 , then w E C; ; this follows from the stminghtforwand extension of Lemma 4.1, to n dimensions. We can now state the following results concerni~~g differentiability. THEOREM5.8 : 8uppose that w E &; @ %' and 9 i s its potential. (i) If M i s of class G: , the Q ,dQ , and 8Q ;2 E2 (ii)I f M ie of cla,ss o:, and m E g2,,then Q , dQ, and 8Q E q 2 n and hence in 0: i f 1 = 1112 - 1 p 0 < p < 1. (iii) I f 42 ie of class 0: and o E $32, then dQ a i ~ d8Q are the potentials of d o and 8 0 , respectively. (iv) I f 44 i s of class C; and uj E c:-' k 2,O < p < 1 then 9 ,dQ and 8 8 E I f 16 2 3 and w E O;-\ thetc Q E CE-l. (v) If M and m a l e of olass Cm OY analytic, then so i s P. I n all case, if we set a = d 9 and /3 = 8Q we have
+
.
+ ,
O F.
,
,
T~EoREM5.9: 8uppose that H i s a havi)&onicfield. (i) I f dZ ;2 C: , them HE CM2a with 1= n/2 - 1 p for any p 0 1 un intero fissato. Esiste : 1) un dominio livzitato T nello spaxio numeric0 complesso a3g-3 (ove
z,
,...,z3g-3 sono le coordinate) omeomorfo ad tlna cells 2) ull domiuio M C (ove 2, r, , ... ,z3g-3 so110 le coordiuate) onleo-
morfo ad una oella e olomorficame~~teequivalente ad nu dominio limittlto. 3) una fuuzione continua a ( t ,r) a valori complessi, - w t m, z E T tale che a (t, z) B olomorfa in z per ogni fissato t, a (t, z) oo per z fissato e I t I m , a ( t i z) f a ( t 2 ,z) se t , t , 4 ) un gruppo 8 di antomorfisnli analitici complessi di M che opera su M senxa pnnti fissi e in modo propriamente disco~ttinuo. 5) u n groppo I' di automorfismi annlitici contplessi di T,propriaw2ente discontinuo (ma non privo di pnnti fissi). 6) nlra applicszione olomorfa z- Z(z) di P nello spnzio di Siege1 delle coppie Z = X i Y di matrici g x g simmetriche X, Y con Y > 0. ed infine 7) u n numero finito di fuazioni meromorfe FJ (2, z) definite su M automorfe rispetto a G. tali che le seguenti conclosioni siano verificate: 8) per ogui z E T le curva y (z): z = a (t,z), - w t m B 121 curve frontiera di un dominio semplicentente cow)ttnssoD (z) 11el p i a ~ ~drlla o variabile Z. 9) UII ptinto (a, z, z3g-3) = ( x , z) B in M se e solo se t E T e z E D (z). 10) ogui elellleuto di G B dell;^ forma
< > g* di T (S,) au 4 (So):
rappreeen-
g* dipelide soltanto da [g] e conserva la distanza di Teichmiiller e l'aualiticittl reale e complessa. I1 gruppo delle rappreser~tazio~~i lecite di T (4) in sB sltrtl denotato con r ( S O ) . Ricorrendo all'uniforlnizzazione mediante gruppi Fuchsiaui si dimostra che : T(8,) B uno spazio mnetrico conlpleto; se il gruppo fondamentale di So B generato in mod0 finito, r ( S o ) Bpropriafnente discontinuo; le funziorli aualitiche reali su T(S,,) separano i puntl. I n base 111 nuovo teorema di uniformizzazione enunciato nel $ 3 si dimostra che : ye So B di prima specie, le funzioni olomorfe su T (So) separano i punti. o) Se So B di tip0 (g, n) scriveremo T(&) = Tg,,, r ( S o )= T',, Questa notazione B giustificata dal fatto che clue qualnnque ~uperficiedi tipo (g, n) so110 quasi-conformemente equivalenti. Porremo Q = 39 - 3 n, ed assumeremo @ 0. La teoria di Teichmiiller [I, 5,13, 141 delle rappresentazioni quasi conformi estremali implica che Tg,,, sia una 2~-cella.'Iuoltre B noto che TgPnB
.
>
+
e Moduli
una varietic analitica con8plessa (cib B stato dimostrato per la prima volta da Ahlfors [a] ; cfr. anche [6, 1 0 , l l , 151). Nel nostro teorema fontl~mer~ti~le, T, I' e M tengono il I~iogo, rispettivamente, di Tg,, , Tsjoe T,,, Le osservltzior~iprecedenti giustifioaoo alouni dei nostri euiiaciati. L1esister~zadelllt rappresentazione descritt,s in (6), (13) segue, ad esempio, dalla formula variaziollale di Rauch [Ill. a) nelllenunciato del nostro teorelna,, T = Tg I I O ~ ltppare come U I I ~ variet8 a~~alitica cornplessa astmtta lntb come un dominie limitato. Questo b un caso pwrticolare di nl1 risultrcto pih generale: Tg.,,8 (olornorficamente equivalente ad) un dominie limitato in Ce. L n dimostrltzione (indicati~sommariamente in [7] b pinttosto complicata. Essa B basata sulla possibilitA di u~iiformizzareogni superficie di Rielna~iliol~iusamediuite grnppi di Schottky, ed involge unc~ltualisi geometrica dettagliata dello spazia di Schottky * di cui Tg b il ricoprimento ~u~iversale. La di~nostrltziorieprocede per induzione su g e su n ; iu tale iuduzione le superficie iperellittiche rivestono un ruolo particolare. e) la rsppresentazione di T,,, nelle forma M, ciob liella forma descritta negli enunciati 8) e 9) e lu, costruzioue del gruppo G avente le proprieth 4), lo), 11) 8 basata sul teorema di uniformiaazione del 5 3. Suppone~ldodi aver compiuto le tappe precedel~ti, IIOII 6 difficile coucludere la dirnostri~zione, ciob eostruire le fuuziorii Fj aver~tila proprieth 7), 14). Fissiamo un insierne di gelreratori Ai, Bj di G (cfr. lo)), e definiamo su ogni #(z) una base di omologia, ehe denotiamo con le stesse lettere. Sia wj il differeuzirble abeliano di prima specie avente periodo ajk su Ak (sicchb, fra I1altro, il periodo di wj su Bkb l'ele~neatoZjk di Z(z)). Sia Qjk il differenziale abeliarro di terza specie su S (z) avente periodi 0 silgli dj e tale che in ogni puuto di S(z) il residiio di Qjk eguagli I'ordiue di wj/wk. L7insieme delle funzioni [w,/wk Qjk/@1j, k, e = 1, 2, g), considerate come fnnzioni 9(z) ha le proprieta ricl~ieste. OS~ERVAZIONX. I1 teorernlt del 5 1 B sfortnuatamente di carattere piuttosto >.Sarebbe utile ltvere espressioni esplicite per i domini e le funzioni ctescritte. 10 esito ad ltffermare che vi sia molta speranzlt di ottenere tali for mule^
.
,
,
...,
g 3. - Un lluovo teorelna d i nniformizzazione. Un gruppo G di trasforr~iazioliidi M6bius sarA chiltmltto quasi Fuchsiano se esiste sulla sferlt iii Riemann una curvlt di Jordan orieiitata y~ tale che y~ sia invaria~iterispetto a G, e questlultimo silt privo di puuti fissi e pro. priamente discontinuo nei domini I ( y G )e E ( y G )rispettivamente interno ed esterno a YG
.
TEOBEMAI. Siritlo 8 , e X2 due superficie di Riemanu. Supponismo che S , e 8%abbiano superficie di ricopri~neuto uuiversitle iperboliclle, e che 8, sia quasi conformemente eqaivalente i~ll'iinmagiue spec~llarej, di S2 In queste ipotesi, esiste un grappo quasi.fuchsiano G t d e c l ~ eI ( y G ) / Gsia conformemente equivalente a 8 , e E (yG)/Ga 8,. OSSI~:RVAZIONE. SB defiuita sostitue~ldociascnaa uniformizzazioue locale 5 sn 8 colt lib suil complessa coiliiigiktii $ Le ipotesi per il teorema 1 Son0 soddisfatte se S , e IE, sono chiese e dello stesso geuere 1. DIMOST~~AZIO Pol~iilluo NE. So = S z . Ne segue che El = SF per un opport,u~lom E B (So).Per ipotesi So = V/Cf0, ove Q B il semipiano soperiore e 6 , B an gruppo fucllsiauo yrivo di p ~ u ~ ~initi. ti P e r t i ~ t ~ tL/GO= o A$,,L esseutlo il sen~ipiauoinferiore. P o l ~ i a ~ lp~(z) o = 0 per 3111z < 0, e definiamo p(x)per 3115 x>O ~nedialltela cotldizione : p(x)&/dz=m. Bisults Ip(z) I ( k < l , e
.
>
Esiste uuo ed nno solo omeomorfismo o, del piano in sB che lascia 0 e 1 illvariituti ed B p-oouforme, ossia B une solozione dell'equazione di Beltrami.
,
Se AE CS, 17equaziotle fuuzioni~leper iinplici~c l ~ ecop ( A (x))B uu automorfismo p-confonne dells sfera di Rieiuana, di guisa che
B uns trasforrnazione di Mobius. Si verifics olie G = w@ Go (or)-' B il groppo quasi-fuchsiaeo richiesto. Indicheremo con I la ra,ppreseiltaziolle natu1:tle di So sn go. Un omeoh morfismo S +So B detto allti-q~~asicoilfor~~ie se pub essere fattorizzato nel h modo segnente : 8 +-S:
