Existence and Partial Regularity in the Calculus of Variations (*).

Ho~-G M:~--C~I:~- (**)

Summary. - We prove existence and partial regular.ity o/ minimizers o/ certain ]unctionals in the caleul~,s o/ variations, under the ~ain assumption that integrands be quasiconvex, and introduce a de]inition o/ generat-guasiminima ( G - Q.minima), and prove some regut(o'ity results ir~ L ~ ]or G - Q.minima as well as for Q-minima.

l . - I n t r o d u c t i o n .

In this paper we consider the problem of existence and partia.1 regular i ty for funetionals in the calculus of variations.

To be definite~ let us consider the lunct ional

Q

where Q is a bounded domain in R'~ u: ~ - ~ R ~, D u - ~ { D ~ u ~ } , ~ = l , . . .~n; ~-----

= ~, ..., 2~, and ]: ~ • ~ • '~ ---> R is a Carath6odary function, namely measurs.ble in x for every (u, 19) and continuous in (u, p) for almost ~ll x G D.

Two principal tasks of the calculus of variations are (a) to prove the existence of minimizers of 57 and (b) ~o s tudy the smoothness of such minimizers.

On ~ho existence problem (a), a suitable condition, t e rmed quasiconvexity, was in t roduced by C. B. MoI~I%EY [12] in a fundamenta l paper in 1952. ] is said to be qu~sieonvex if for every ~oG [2, UoG R ~v, peg R "~' and for every r G C~(.O, R z') we have

(3 .z) /(~o, u0, p0)< f/(~,0, uo, po + Dr az t9

~Iorrey showed tha t if ] is qnasiconvex and if certain cont inui ty and growth hypo- theses are st~tisfied, then for various b o u n d a r y - v a h e problems there exist minimizers

(*) Entrat~ in Redazione il 10 settembre 1986. (**) This work was done in Na.nkai University. !ndirizzo dell'A. : Department of ~iathematics~ Zhejiang University, Hangzhou, P.R. China,

312 tIo~G Mx~-C~tu~: Existence and partial regularity, etc.

for 37. Morrey's remarkable existence theorem fails to apply directly to nonlinear elasticity because his growth conditions are too st.ringent. J. BA~L in [2], [3] has modified Morrey's ideas and derived important existence assertions for several prob- lems in the nonlinear elasticity.

Progress concerning the regularity problem (b) has been to date less definitive. In a recent paper [9], M. GIAqVI~TA and E. GrusTI (using direct methods) have shown that the regularit.y of the first derivatives of minimizers of ~ under the main assumption that ] be strictly convex in p. In a more recent paper [12], L. C. EVAns has proved that the regularity of ~he first derivatives of minimizers of the functional

(1.3) ~o(U; O) =jfo(Du) dx O

under the principal assumptions that integr~nd ]o is uniformly strictly quasiconvex, and does not depend on x and u. During ~ leetur~ in ~ank~i institute of Mathe- matics, professor E. GlVSTI mentioned the following problem:

Whether the result of Evans is still true for the functional (1.1) without the coercivity assumption of the form

(1A) ](x, u, p ) > 2Ip [ . . . . a (2 > O, a > O, p ~ R "~)

and he gave the following example which does not satisfy the coercivity assump- tion (1.4) for any 2 > 0 , m = 2 . Let:

](x, u, p) A~.Z(x, ~ = u)p~pz 4 - L det (p) (L > 0, n = N = 2)

where the coefficient A~iZ are continuous in ~ • N and satisfy the condition:

~ ) ~ , ~ > ~ [ ~ i 2 , V$ ~ W'~'; v > 0 ,

gad L is & large enough constant.

A brief outline o f this paper is as follows. Ill w 2, we defii~e ]o(P) to be strictly quasieonvex in p at zero, and extend Q-minima (see [10]) to G - Q-minima, and replace the eocrcivity assumption (1.4) by (H4) and finally obtain L~-est.imgtes for G - Q-minima. In w 3, we prove the existence of minimizers of the functional (1.1). In w 4, we give a positive answer to Giusti's problem under additional hypotheses on the function ](x, u, p). Roughly speaking, we assume that ] is twice differentiable mid uniformly strictly quasieonvex ill p, gird HSlder-eontinuous in (x~ u).

Notation. - W e adopt the notion of GIAQUI~TA'S book [6].

HONG ~IN-CHu~: Existence and partial regularity, etc. 313

2. - General quas imin inm.

First we consider the functional -

(2.1) 5o(U, Q) = f lo(Du) dx D

where ]o(P): R'~"-~R. Let us assume 2 ~ < m < c~ and ]o(P) satisfies the following conditions:

(H1) ]o(P) is twice differentiable, and

I/,,,(p)i-~L(1 § tplm-~).

]o is strictly quasieonvex at O, i.e. there is a constant ~ > 0, such tha t

O~ O~

for all smooth, bounded open domain 01c R ", and all f e ~(01; R -~') with r -= 0

on ~0~.

DEFINITION 2.1. - ~ We say tha t u ~ H~;~(D, RaY)is a general-quasiminima (G--Q- minima) for the functional (2.1) in D if for every open set A coD and for every v ~ ~oo~r'a'(o,~., R N) with v-----u outside A we have

�9 ~o(u, A)<Qo~o(V, A)-t- Gf(lDv['~+ 1) dx A

where Q~>I, G~>O are constants. Since 5o is an integral ftmetional, u is G - Q-minima for 5re if and only if for

every r ~ H~,.~(D, R ~v) with supp r = K c .(2 we have

Yo(~, K)<-<OYo(~, + r K) + Gf(ID(u + r + 1) (2.~) dx. K

The following two lemmas are very useful in what follows. They can be fourtd in [6].

LE~fA 2.2. - Let ](t) be a bounded non-negative function in [To~ T~], such tha t for every s, t, To < t < s < T~:

/(t)<A(,s.-- t)-~+ B + 0i(8)

with A~ ~ B~ 0 non-uegativ% and 0 < 1,

314 I - Io :~ M ~ - C g u N : Existence and partial regularity~ etc.

Then there exists a constan~ c~ depending only on ~ and 0~ such tha.t for every o7

(2.3) ] (~)<c[A(R- - ~)-~-~ B] .

L E n A 2.3. - (Reverse H61der inequali ty) .

Le t B be an n-ball, let ]~ L~or , q > 1, ] > 0 and g SLiceS(B), (5 > O. t h a t

Suppose

for each x o e B and each R < dist (xo, ~B), then there is ~ small cons tan t So> 0,

] e L~oc(B) and p e [q, q ~- So) and every ball B.~cc B, such t ha t

(~.~) (f]d,'~13) 1[1) ~e(~]qdx)l[q4 - C(~Y(~X) I]~0 ~r12 B~ B~

where e and so are posit ive cons tan t s dependfilg o n bi q, n, a. ~-ow we can prove the Firs t Caccioppoli 's inequal i ty

Ttt~0RE~I 2.4. - Le t u be a G - Q-minima o f ~o with ]o satisfying (HI) , (H2),

then exists ~ cons tant c such ~h~t

f f (2.5) (~:Dup-~ 1) "/~ dx< ( R ~ ) , , [u - -K[ '~dx 4- cIB(y, R)i B(v,q) ~(v,~)

for all B(y, R) c Q, K e R N.

P ~ oor . - We m a y assume y = 0. Le t R/2 < t < s < R and choose ~ e C~ (f2~ R) sat isfying

r t --= ] on B(t)l~ ~ = 0 C

0 <,~7 < 1 , (Dn s - - t

on 9\B(s)

Define

- - ~ ( u - - K ) , ~ - ~ ( 1 - - ~ ) ( u - - K )

then

(2.6) u - K = ~ -{- ~ , Du = .Dq7 -~ D~o .

Ho~6- MIzx-=C~u~-: Existenee and partial regularity, etc. 315

Since u is a G - - Q - m i n i m a , we have

(2.7) io(Du) ax<Q ]o(Du-- Dq~) ~ ~ -- , = B8 B~ Bs

= Qf]o(D~f)dx @ Gf(t.D~fl', @ 1) dx. Bs B8

Condit, ions (H1), (H2)imply

(2.s) B~ Bs Bs

@o(.u) lo.(.u) + f + ) Bs .Bs Bs

Bs B8 B~ Bs

<~f(Iowi,~ + ~) dx + 0f(IDui'~-~ + I)[Dw I dx< B~ Bs

<ef IDv, i" d~ + @Dul,,-~IDwI a~ + ~iB,I . B~ Bs

For ~ = 0 on Bt and DyJ = ( 1 - - ~ ] ) D u - - (u-- K) D~],

�9 o [DvJI*~<e(IDut~+ I n - K p I D v l '~) or~ B ( s ) \ B ( t ) ,

IDyJ]lDulm-~<~eIDul ~@ c(e)lDw[ ~ , e > 0 .

Let s = ~/2e, thus (28) yields

f f e fIu--Kl"dx+c!B.t IDu!~§ l l ) " /~dx<e (IDut~§ l)"*/'dx § ( ,~_t) ~ Bt B , \ B ~ B~

Adding to bo th sides, e t imes the quan t i ty on the left, and dividing b y e -~ 1, we obta in

f f e (lDui ~ + 1 ),~/~ dx + (~ --t) ,~ tu - - KI ~ dx + e~lB, l . (IDuI~ + 1)~I, d x <

.B~ Bt Bs

B y using l emma 2.2 we obta in

(tDul~@l)~t*dx<v~ R~_~)~ lu--g]mdx@ IBm] . q.e.d.

We can now prove the L*-est imates for G - Q-minima of 2v0.

316 HONG _~hN-C~uN: Existence and partial regularity, etc.

THE01CEI~[ 2.5. - Let u a G - Q-minima of 3~o with ]o sat isfying (E l ) , (H2), then r r l ,p to RN). Moreover there is a cons tant So> 0 for p ~ [m, m ~-so), such t ha t u ~ l o e W . ,

there exists Re > 0, cons tant c > 0 such t ha t for every ball B~ c .(2 wi th 2R < Re we

have

,,, I~ (,o,,,,-.~ ~),,, q" ~I~ o,,,.+ ~.,,,<q "~ .

PROOF. - Le t K = u~.,~ ~- 1/IB~lfu dx in theorem 2A where B,~ : B(xo, R), we BR

have f ( u - - K ) = 0. Let ~ = R/2 and by u~ing Soblev-Poincare inequali-~y, f rom BR

(2.5) we obtain

{~r (,~ <,)-,. ,,../ ~. / ( f iD,,, .... 4"" " '+ f<,4~ BR/2 BR BR

we can now use l emma 2.3 to obta in

{ f (IDui~ ~ l . )" i~d:c} ' /P<-c{f(IDui~-f - B~ B ~

l~l" f 1) ~'~/~ dx / -t- c dfx,< B~/~

I f , " - < c (tDul ~ -~ 1) "/'2 dx I . q.e.d.

3lore generally~ we can consider the same problem for the funct ional (1.1), we

assume t h a t J satisfies the following eondRion:

(H3) There exists a certain funct ion ]o(P) satisfying conditions (H1), (H2), such tha t

t ~ ( P ) - - g < ] ( x , u, p) < ~IPl" -I- g

where g > 0~ ~ > 0~ and m ~ 2 are constants .

- ~1"~to R ~') is called a G - Q-minima for the DEFINITION 2.6. A function u ~ ~loo ~ funct ional (1.1) if ] s~tisfies (H3) and if there exist constants Q ~ ] , G > 0~ such t h a t

(2.10) X(u; ~l)<Q~-(v; A) + Gf (IDvh + 1) dx A

~1,~/O R~V) with v = u outside A. for every open set A c c / 2 and for every v ~ ~1or ~-~,

H o ~ MIN-CI~u~-: Existence a~d partial regularity, etc. 317

RE~AI~K 2.7. - (i) I f f satisfies (Ha) with ]o(P) = ),[P[~, then the ( 7 - Q-minima

of the funct ional (1.1) is a Q-minima of the funct ional (1.1).

H~,',,tO R x) (ii) I f ](x, u, p) s~tisfies condition (H3) and we suppose t ha t u e ioo w-, is a G - - Q-minima for the func t iona l (1.1), then u is also a G - - Q-minima for the

funct ional .To(U; .(2) =- f A(D'*O dx.

Since condition (H1) implies

[D/o(p)l<(]- + IpP-~), [to(p)]<~(~ + tP["),

then by choosing 1, g large enough we can prove (ii).

F r o m r e m a r k 2.'7 (ii) we have

TH~oo~m~r 2.8. - Le t u be a G - - Q - m i n i m a for the funct ional (1.1) in ~ , a.nd

assume t h a t (H3) holds. I f N > I ~ then there exists an exponent q > m such tha t 1,q R N ) u e HG(~, . And moreover (2.9) is t rue.

We conclude this section with the example t ha t we halve s ta teed in section 1.

EXAMPLE 2.9. -- Le t us consider the funct ional

5(~; ~) =fl(x, ~(x), Du(z)) dx

where /(z, u, p) = A ~ [ ( x , ~ J ' u)p~pa T L. det (p). A~ ~ are continuous functions in ~ x R ~'

satisfying the ell ipticity condit ions:

A~'~S~ ~ > v [ ~ [ 2 v ~ e R ' ~ ; ~ , > o , n = N = 2 ~;j ~ ,

By choosing ]o(P)= vtPP+ L det (p), i t is easy to check that ]o(P) satisfies (H1),

(H2), and J(x, u, p) satisfies (H3).

3. - E x i s t e n c e o f a m i n i m a .

In this section, we shall derive existence theorem b y modify ing Morrey ' growth

condition. Here we use the me thod given by P. MAROELLI~I and C. S~ORDO~E [15]. Le t us rooM1 the following var ia t ional principle in EI;ELA~1) [4].

Tn-~o~n~ 3.1. - Le t (V, d) be a complete metr ic space, and let Jr: V -+ [a, @ co]

be a lower semieont inuous functional , not identicMly + 0% let V > 0 and v ~ V

be such that

~-(v) < i n f 5= / - r/ . jr

318 Ho~ ~L~-CHt~: Existence and partial regularity, etc.

Then there exists f i e V~ with d(5, v)~<l such t, ha t 5 ( f i ) < 5(v), a.nd moreover

(3.1) 5~(~) < 5(w) + Vd(u, w)

for every w e V.

R E H A n K . - ~Vriting u-}d instead of d we can also conclude the existence of u ~ V with d(u, v) < U}, minimizing 5(w) @ u~d(u, w) in partieulsx if {vT0} is o mini- mizing sequence in V~ the corresponding sequence {m~} is also minimizing, and d(u,~, v~) ~ O.

We have ~he following semicontinuity result [5].

THEOBE3I 3.2. - Let f be a quasi-convex function sa.tisfying

(3.2) IX(x, u,p)I<ylpl" @ g

with m~>l~ g~ # > 0. Then the functional

5T(u; s = f ](x, U, DU) dx ~Q

is sequeutially lower semi-continuous in the weak topology of ~'q Hior , R ~) for every q > m .

L E ~ . ~ 3.3. - Le t ]o(P) satisfies (HI), (H2), then ]o(P)- etPl m also satisfies (H1)~ (H2) for e > 0 small enough.

P~ooF. - Le t ]~(p) = /o (p ) - elpl ~. Since re(P) satisfies (H1), (H2), we h~ve

ID~]o(p)I<c(1 + IpI ~-~)

and there exists v > 0, for R > O, xoe R '~, r ~ C~(B(xo, R), R ~v) such tha t

fs~ + B(zo,R) B(xo,~) B(x0,i~)

Then we have

and

ID2]~(p)] <c(1 -[- IpI "-2)

HONG MI~,C~u>-: Existe~tce and partial regularity, etc. 319

By choosing s < v/2, we obta,in

So ]~(p) satisfies (H1), (H2). q.e.d. We can now obtain existence theorem.

TItEORE~ 3.4. -- Let ] be a quasi-conve~ funct ion satisfying (H3), and ]o(p) satisfies (H1), (H2) with ]o(p)>0, then

,~(u; ~q) =i'/(x, u, Du) dx

at tains its minimum in the class V---- {v: ~ H~"*(~, R'V)}. Moreover the minimizer 1,q u ~ Hloo(~9 , R I7) for some q > m.

P~ooF. - The class V is a complete metr ic space with distance

d(u, v) =[fDu-- Dv[ dx. D

B y using Fa tou lemma and ]o(p)>0, it is obviously seen tha t the funct ional 5- is d-lower-semieontinuous in V. Le t {v~} be a minimizing scquene% and let {uk} be the corresponding (minimizing) sequence given by theorem 3.1

97(uk; ~ ) < 5 ( w ; 9 ) + ~7~]Duk-- Dw I dx . D

I f K = supp (u~-- w) c D, we have

; K) < 5(w; K) + ~ [Du~-- Dw I (3 93~ 5(u~ dx. K

Oa the other hand, using (H3), we have

K K

and

K K K K

where A~, B are constants, and notice tha t a < e a . ~ + c(e), m > 1, ~ > 0, a > 0. Thus f rom (3) we obtain

K K K K

320 I~o~G ~II~*-CHu~: Existe~ce and partial regularity, etc.

If we select e small enough, we h~ve

(3.~) f[?o(Du~)- e[Du~:]-~] dx<Qf[A(Dw)- s[Du~i'] dx -~- aj'(IDwl ~, H- 1) dx K K K

where Q, G are constants independent of I:. Oll the other hand, we can use (H1), (H2) to obtain

(3.5) D D .(2 .('2

From (3.5) we can conclude tha t the sequence {u~} is bounded in H~,,~(~, R~). Since m ~ 1, we can suppose tha t u converges weakly to some funct ion u e H~,~(~, R~).

Using (3.4) we know tha t u~r is a G - Q-minima for the functional ~ - ~ f l d D w ) dx.

Frome lcmma 3.3 and theorem 2.5, we can conclude tha t the sequence {u~:} is equi- bounded in n~or R N) for some q ~ m, and therefore u~ converges to u weakly

in *~lor176 R~). F rom theorem 3.2 we conclude tha t u gives the required minimum.

I:~EMAI~K 3.5. -- I:[ Uoe HI,m(sQ, R ~,') a n d V = (v e Hl,m(ff2, R~'): v - - u o e H~'m(~Q, R~V)},

we can write ]l(x, u, p) : ](x, u - uo, p - Duo) instead of ](x, u, p). Assume/~(x, u, p) satisfies the conditions of theorem 3.4, then the conclusions of theorem 3.4 holds.

In a recent paper [12], L. C. EVANS considered the problem of regular i ty for the functional (2.1) under the main assumption tha t 1o is uniformly str ict ly quasi- convex. Bet he still need the coercivity assumption (]o(P)>~ ]P[~'~) for the problem

of existence by using ACERBI and Fvsco ' s results [1], and Z~C[ORREY'S existence theo- rem [14]. Here applying our theorem 3.5 to this problem~ we can prove the existence of minimizers of the functional (2.1) wi thout the coercivity assumption.

4. - P a r t i a l r e g u l a r i t y .

In this section, we shall extend Evans ' results to more general case. Let us consider the functional

(4.1) 5(u; ~9) = / / ( x , u, Du) dx . Q

First let m ~ 2 and assume the following assumptions:

(H3) re(P) -- g•1(x, u, p)<~1p{ ~ -~ g ; ~, g > 0 .

where re(p) satisfies (H1), (H2).

I-Io_~G MIN-CtIu~: Existence and partial regularity, etc. 321

(H4) For every (x, u) ~ D x R -~, ](x, u, p) is twice differentiable in p, uniformly respect ~o (x, u), and

(4.3) I.G(x,u,p)!<L, L > 0 .

(H5)

(4.4)

Suppose ] is uniformly str ict ly quasi-convex; ha, rely tha t there exists ~ ~ 0, for every Xoe Y2, a ball B c R ' , u0e R -~, poe R "~x" and for every r ~ C~(B, R ~')

f(~o, u0, po)!BI § ~f lDr ~ dx< fl(~o, Uo, Po § De(x)) dx . B

(H6) For every p ~ R ~N the functioi1 (1 § IpP)-V(x, u, p) is eontinuou~ in (x, u) u~iformly with respect to p, in other words there exists a bounded con- t iuuous concave increasing funct ion o)(t) with ~o(0)= 0 such tha t

(4.5) It(x, u , p ) - l(y, v, p)l< (]- § IpP)~o(Ix- y p § I ~ - ~1 ~) �9

The next theorem is an essen'fial step in the theory of part ial regularity.

THEOICE)I 4.1. -- (Second Caccioppoli's inequali ty). Le t u be a minimizer of :F(u; ~) , let (H3), (H4), (H5), (H6) hold and let P(x)

p o ( x - Xo) § a be a polynomial of degree 1, then

f c f , u -Ppdx+ (4.6) IDu--pol~dx< ( R - - e ) 2 Be JBR

c t (1 + IDup § pg)o~(tx --,~o? + 41~ - - uol' + po~J~ - - Xoi") d~ § .BR

where p~=~Du dx, a = %~ =~u dx. Brt BR

P R O O F . - Let fl be a cut-off funct ion (as in theorem 2.4) and le~

q~ ---- V ( u - P ) , y~ = (1 -- ~/)(u-- P)

then

U -- P --- ~ § y~ , D~t -- p~ --~ Dcp § DyJ .

Since supp ~ c Bs c tP, then

(4.7) /(Xo, %.R, Po)lB,! + ~,f lDq~l" d.~< f/(Xo, %.R, po + _D~) dx =

=f](Xo, ux~ D u - Dye) dx< .Be

< f/(Xo, %,,, Du) ax-f/~(Xo, %,,, D~) l)W ax § Lf lDwp ax = B8 Bs t~

322 Ho~G MI.~-C~v~-: Ex,;stence and partial regularity, etc.

=f / (x , u, Du) (lx 4-f[/(x~ u~,,R. D u ) - /(x, u, Du)] d x -

B~ B~

Since u is a minimizer of ~ then

~-(u, B < ) < ~ ( u - 9~, B ) .

By using (4.5), we h~ve

(4.8) Y(xo, u~,,., p)lB.I + vf iDqo P dx<fl(x, u - % D u - D?) dx + B8 17s

Bs t~s Bs

-~f/(x~ u~o,R, Du Dr) ax 4-lEt(x, u - V, nu -- D r ) - /(xo, u~,,,., Du -- Dr) ] dx ~- Bs B,~

xo,R] ]

Bs Bs Bs

Since

u - ~ = u - v [ u - p ~ - x~ - u . , . ] , tu - ~ - % , . t < ~ l u - % , . I + [pol]x - Xol

and using condition (H6), we have

(4.9) f I/(x, u - ~, D u - Dq~) - - / (Xo , u~,,R, D u - Dq~)l dx< Bs <fo)(tx- ~o1~+ ~lu- %,.I~ + p~i~- x~176 + IDu- D~I ~) e~<

B~

<f(1 @ 2pX @ 91DyJi "2) co(ix -- xot ~ @ 41u -- u~.,.l 2 @ p~lx -- xol ~) dx Bs

thus we have

(4 .1o) f /(x., u~~ p~ dx § ,,f lD~i 2 dx< B8 Bs

< f/(xo, U~o.., Po @ D~o) dx + j ( 1 @ tDut2)o~(tx - xop @ [u-- U~.o,.l ~) dx @ B8 B~

@f(1 @ 2po2@ 21D~ol~)w([x -- xoI 2 @ 4]u-- u<,, ~t~ @ p~Ix-- Xol ~) dx-- B~

B~ 1~

HO_XG 5[IN-CI-IUN: Existe~we and partial regularity, etc. 323

<ft(Xo, %,~, po) d~ +f[],(~o, %.,,, po) -/,(~o, %,, , D~,)] 9V a~ + B~ Bs

+ of(l + ID~i~+ P~o + tDvl2)~(Ix-- Xo[ ~ + 41u-- %,.I~ + PNI~-- Xolg) ax + B,

+ 2L f i D e ? ~ B,

and using (H4) , we have

13~ B~ Bs

+ of(1 + p X + IDul")'~([ x - Xol2+ 4 t u - u . . ~ l ~ + p X l x - Xop) dx , B.

and

Bs B s ~ B t B s ~ B t

+ of(1 @PN+ [Dul~)~ x- Xol~@ 41u-- u~,,.]~+ P~ol x - x~ ~) dx. B,

Since

!D~i <elDu --Pol + - - - - - C

(8 - t) ~'~ - f i ,

we h~ve

ts--~)d B~ B ~ B t B~

f § (1 @ !Du]~@p~)og(,.~)dx. B,

We (~ fill the hole ~) b y adding e~flDu- po[ ~ dx to bo th sides and thus obtaining Bt

c~ f ID u Bt Bs

We can ~pply l emmz 2.2 to derive

- -Poj - - d.~ -~- (s B~

f 2 (,.,~., ~ + o (1 + iv~,/~ + po)~O, , r B~

f IDu--Pol~dX< (R e )~ f lu--Pl2dx@e f Bo 2R B~

(1 T [Dui 2 + po~)(,4 --) d~:. q.e.d.

i~m1Al~l~ 4.2. - Suppose fur ther tha.t for some Mo and some R < R(Mo)<I~ we

h~ve !Po[ < Mo. Now let Q : R/2 in (4.6), noticing- t ha t f ( u - P) dx = 0 r and using B~

t to~G M~N-CHu:~: Existence and partial regularity, etc.

Poiuca.re-Soblev inequaligy, we have

BR/2 BR BR

let us recall theorem 2.8 (Z~-estimate), there exists some co> 0 such tha t u e 1,q e Hloc(D , R ~) for q ~ [2, 2 -~- co). Thus by using lemma 2.3, we can obtain

-~a"

T~E0~EY~ 4.3. - Suppose tha t ] satisfies (H3), ..., (H6) wi~h w(t)<<At-, a > O, and 1,2 u eH~or R ) is a minimizer of 97(u; f)), then there exists an open subset

f2or f2 such tha t ]f~\f)o[ ---- 0 and D u ~ C~(~o, R -~') for some 0 < cr < 1.

P~ooF. - We shall est imate

U(xo, q) -= ~ [Du -- (Du)~,,q[~ dx ; Be. -- B(xo, o~).

Let xoe f2, R < dist (x0, ~Q) and let

(4.16) I(v; B~) "-----f ]~s ~i(xo, u,.,~, po) D~,v ~ .D~vJ dx . B,n

Lot v eH~,2(B,) be the minimum for I(v; BR) with boundary v~lues u(x) on 8B~,

thus v satisfies tho Euler equat ion:

f /w~ Po) D~ v~ Daq# dx = 0 Vq~ e H~'2(B~, R ~) ~ (Xo ] R~ o

fir

We recall f rom MO•REu [14; Thin. 4.4.3, Thm. 4.4.1] tha t condition (H5) implies

the strong Legendre-Hadamard condition:

for ~ e R ~ , N e R ~. Thus linear elliptic P .D.E. theory asserts v is smooth and provides the following

estimations (see [6], [16]):

(a) f Dv dx = f Du dx = po

I-IoNG ~Iz~-CHu~': Existence and partial regularity, etc. 325

(b) f] .OV--po]~dx<vf[Du--po[~d~'; ( q > 2 )

B~ B~

(c)

Be BR

(d) f !Dv l~dx<vf !Du~dx; ( q > 2 ) .

B~ B~

Combing (b) with (c), we obtain-

(4.1.8) f iDu--{~u}o.i~ dx<e(-~)'~§ !.u--{~)u},!~ dx § liD(u--v)i~ ax. Be B.~ BR

Let

q~ = u - v - [ u - p o ( x - xo)] - Iv - p~(x - xo)]

and from (4.17) we can obtain

Bn BR .B,~ ,

= fJ , , ( xo , U~o,R, p o ) [ ( D u - p o ) ( D u - Po) -- (Dv -- po)(Dv -- Po)] d x < BR

< f [J~,,(xo, ux.,R, po)(Du - po)(Du - Po) -- 2/(xo, u~.,~, Du)] dx ~- BR

+ f[u/(Xo, u~.,R, Dv) - f,~(xo, u~,,R, po)(Dv -- po)(Dv -- po)] dx + BR

+ 2f[l(x,., D~)-/(~, v, D~)] d~ + 2f[/(Xo, %,~, D~)- BR Bz~

- - ](x, u, Du)] dx -~- 2 f [l(x , v, Dv) - - ](xo, u,~.,R, Dv)] dx -~ Bn

-- L + z~ + z~-t- I~ + z. .

Notice tha t I a < 0 , since u is a minimizer. And by using (H6), we have

(4.20) z ,+ zo<f(1 + I-ul')o,(Ix- .oil+ Iu- %,~i'),~x + BR

326 HO~G MI~-C~[u~-: Existence and partial regularity, etc.

Since ] is twice differenti~ble in p, we h~ve

](Xo, u~.,~, p) -= ](xo, u~,,a, Po) § ]~(Xo, u~~ Po)(P -- Po) § 1

§ - - t)/~,(xo, U~o,R, Po § t(p - - Po))(P - - Po)(P - - Po) dt 0

aad using estimation (a), we have

(4.21) 1

z~ + r~= 2~ f(1 - t ) [ t . (po)- i .(po + t (D~ = po))](D~-- ~,o)(D~-- J,o)at ax + .B.~ 0

1

§ 2 f f (1 -- t)[]~(po § t(Dv -- Po)) -- ]~(po)](Vv -- po)(Dv -- Po) dt dx . BR 0

(H4) implies tha t there exists a bounded continuous concave increasing function

~Oo(t) with ~o(0) = 0 such tha t

[&(Xo, u~.,~, po)- G(Xo, %, . , p)t < ~o(Ip - poP) �9

From (4.19), (4.20) and (4.21)~ we obtain

(4.22) flD(~- ~)l~e~<f~o(ID~-por')lD~ -pol~ex + BR BR

+ fmo(lDv - pol~)lDv - - pol ~ dx § f (1 § IDul~)co(lx - Xot ~ § l u - u~.,~[ ~) dx + B~ BR

+f(~ + ID~l~)~(Ix- Xol~+ [u- ~,I~+ I~- ~o,~I ~) d~ �9 Z~+Z,+I~+ f~. �9 B / ~

By using theorem 2.8, remark 4.2, the boundedness of w, we, and the reverse tt61der inequali ty (lemma 2.3), we can estima.te the terms on the right-hand side of (4.22):

BaR

• § z,<c f B~.R

IDu --poi~ dx [ f oJ~/~-~(iDv -- po'2)] dx + B2R

B~e B~.n

dx (q > 2) .

(For the details of the derivation see 5L GIAQUI~TA and G. MODICA'S paper [11].)

Ho~-~ MIN-CHu~ : Existence and partial regularity, etc. 327

Let

= r R , M(xo, R ) = 1 -b i(Du)~o,RI @ U~(xo, R) , r < ~ .

Since O~o is concave, and using Jensen's inequality, from (4.18), ..., (~.24) we have

(4.25) V(Xo, ~.R)<~{~ v(x0, n) + R'-(~:-'/r R) -~ ~o(oU(Xo, R))~(xo, R)}

where c is a constant , q > 2.

The following proceeds of proof can be done in a s tandard way (see [6], [8], [ll]). q.;.d.

After finishing this paper we learned from Giusti tha t Giaquinta and Modiea

got the similar results to section 4 for m~> 2. But they can not drop the eoereivity

assumption (1.4), so they can not deal with cxa, mple 2.9 given by Giusti. And we

found tha t if combing our results with theirs~ our theorem 4.3 for m ) 2 is still true.

THEO]~E~I 4.5. - I f we replace their assumption (H.1) in GIAQUI~T.~ ~nd MODIOA'S

pa, per [11] by our assumption (H3), ~nd let their assumptions (H2), ..., (H6) hold,

= ~r l 'wo R ~) be u minimizer of the functional (1.1), then there exists an

open ~et sg0c D .~nd ]D\Sgol ~ 0 such tha t D u e C~(~o, R ~') for some 0 < ~ < 1.

Acknowledgment. - I am very grateful to professor E~RlCO GIUSTI for interesting

me in this problem and for many comments and suggestions, mid to professor CHEN YA-ZHE for his valuable advice.

REFERENCES

[1] E. Acn~BI - N. Fusco, Semieontin~dty problems in the calculus o/ variations, to appear in Arch. Rat. Mech. Soc. Anal.

[2] J. M. BALL, Convexity conditions and existence theorems in the non.linear elasticity, Arch. Rat. Mech. Soc. Anal., 63 (1977), pp. 337-403.

[3] J. M. BALL, Strict convexity, strong ellipticity, and regularity in the calculus o/ variatio~s, Math. Prec. Camb. Phil. Soc., 87 (1980), pp. 501-513.

[4] I. EK~,LA~D, Nonconvex mi~dmization problems, Bull. Amer. Math. Soc., t. 1 (1979), pp. 443-474.

[5] N. Fusco, Quasi-convessit~ e semicontinuit~ per integvali multipli di ordine superiore, Ricerche di ~at., t. 29 (1980), pp. 307-323.

[6] M. GIAQUINTA, Multiple integrals in the calculus o/variations a~d nonlinear elliptic systems, Annals of Math., Studies, t. 134, Princeton University Press, 1983.

[7] ~ . GL4-QUINTA - E. GIUSTI, ~Yonlinear elliptic systems with quadratic growth, Manuscripta Math., 24 (1978), pp. 323-349.

[8] M. GIAQUIN~A - E. GIUSTr, On the regularity o/the mi~ima o/ variational integrals~ Acta. 3iath., 148 (1982), pp. 31-46.

328 ItO~G MI_~-C~[u~: Existence and partial regularity, etc.

[9] M. GIAQUI~'rA - E. GIU~TI, On the regularity o] the minima o] rwn.di]]ere,~tiable ]unetionats, Inventiones Math., 72 (1983), pp. 79-107.

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[11] M. GIAQUINTA - G. MODICA, Partial regularity o] mini~dzers o] quasicor~vex integrals, preprint.

[12] C. L. EvAns, Quasiconvexity and partial reg~darity in the calculus o/variations.s, to appear. [13] C. B. MORR]~Y JR., Quasieovexity and lower se~icontinuity o] multiple integrals, Pac. J.

Math., 2 (1952), pp. 25-53. [14] C. B. MoRmon- JR., Mul~ple integrals in the calculus o] variations, Spinger, New York,

1966. [15] P. MARCELLINI - C. SBOP~DONE, On the existence o] minima o] multiple integral o] the cal.

eulus o] variations, J. Math. Pure Appl., t. 62 (1983), pp. 1-9. [16] E. GIcsTI, Lecture Notes in Nank~ Institute of Mathematics. [17] W~y LA~CHE~G, Existence, uniqueness and regularity o] the mini,~bizer o/ a certain ]unc-

tio~a~, to ~ppear.