( 3 ) k? (*. ^)l 2 s M <! + l«l 2 ) (f 2)/2 {bl + /(*)} \a=l*=1 / F. Leonetti ON THE REGULARITY OF WEAK SOLUTIONS TO NONLINEAR ELLIPTIC SYSTEMS^) - PDF

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REND. SEM. MAT. UNIVERS. POLITEGN. TORINO Vol. 47, 1 (1989) F. Leonetti ON THE REGULARITY OF WEAK SOLUTIONS TO NONLINEAR ELLIPTIC SYSTEMS^) Abstract. We consider a vector-valued weak solution it : Q *

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REND. SEM. MAT. UNIVERS. POLITEGN. TORINO Vol. 47, 1 (1989) F. Leonetti ON THE REGULARITY OF WEAK SOLUTIONS TO NONLINEAR ELLIPTIC SYSTEMS^) Abstract. We consider a vector-valued weak solution it : Q * R of the nonlinear system di\a(x } u(x), Du(x)) = 0, x G 0. Under suitable assumptions on the vector-valued function l a\ we prove that the gradient Du belongs to some Morrey space; in the two-dimensional case, that is Q C R 2, such a result allows us to claim the holder-continuity of the function u. INTRODUCTION Let Q be an open subset of R n yn 2;\etu:Q-+R N be a weak solution of the nonlinear system of N partial differential equations *» (1) -^A{af(^,«W,^w(^))}=0 x Q.:'..;' a=l,---,jv where ), = - and Du is the gradient of u. We assume the following OXi hyphotheses: there exist positive constants q } v,m such that: q 2 and N n (2) ^^af(x, u,p) P j, kl + l«2 ) ( '- 2)/2 lp 2 (ellipticity) a=li=l / N n \ l l 2 ( 3 ) k? (*. ^)l 2 s M ! + l«l 2 ) (f 2)/2 {bl + /(*)} \a=l*=1 / (* AMS (MOS) 1980 Classification: 35J60, 35D10. 40 for almost every x G ft, for every it R N, for every p R nn. In this paper we prove a regularity theorem concerning weak solutions to elliptic systems (1) verifying (2) (3). On this subject we quote [7], [3], [4], [2, p.150], [5], [1]. Before presenting all the details of our theorem, we want to show where systems (1) (2) (3) come from. Let us consider the integral functional (4) G(«)= [ {\ + \Dv(x)\ 2 ) q l 2 dx where v : ft» IR m. Let us assume ft bounded and connected; if v belongs to the Sobolev space H 1 ' g (Q;lR m ) then the functional G attains its minimum on the convex set K = {v 6 //^(ftjht ) : v - v 6 i/ 0 1,9 (Q;lR m )} Let us denote v the unique minimizer of G on K. Such a v satisfies the Euler equation / ( 5 ) / E /JW*)) Di* h (*)d* = 0 V* i# f (O; BT) where g( ) = (1 + 2 ) / 2, { = {#},i = l,...,n;/i = l,...,m. By means of difference quotient tecnique [2, p.43-46], [8], we get «e^(q); (i + \DvW-v 2 \Dv\ 2 ei^itt); (i + i?t, 2 ) '- 2 2 z^ 2 eluq) so, if f Co (Q;IR rn ), we insert $ = D s f in (5) and we integrate by parts: we get (6) KfeufeWKl y t ec^(q]r m ) } v* = i,...,n. We set -u = Dv, a^p) = E?=i EZLi ^TWP?,; then equations (6) are the weak formulation of 7) E n._ Di{a % hs(u(x) } Du(x))} 0 for a.e. x 6 ft; V/J = 1,, m\ Vs = 1,, n. This is the system of type (1) with N = nrn equations, verifying hypotheses (2) (3), provided v - q,m = rinq(q - 1), f(x) = 0. NOTATIONS AND ASSUMPTIONS Now we state our theorem. In order to write everything in a more compact way, we use the vector notation with scalar product instead of components and explicit summations. ft is an open subset of IR n, n 2; a = {af}., t = i,, n; a = 1,, N\ N 1; a : QxR N xu nn l\v N is a Garatheodory vector-valued function, that is (x, u,p) a(x, u,p) is measurable with respect to x and continuous with respect to (u,p)\ we set \v\ 2 = ELi(2/0 2 for everv V = (j/v V k ) #* for every integer k 1; we set V(u) = (1-f x/ 2 ) l^2 for every iz R N. We assume that (0.0) q 2 (0.1) 3i/ 0 : (a(x, «,p) p) i/v? ~ 2 («) p 2 (0.2) 3M 0 : a(z,u,p) MV - 2 (u){ p + /(x)} for almost every x G ft, for every u G Ht^, for every p em nn, scalar product in IR n. 41 where ( ) is the We say that a vector-valued function u : ft IR^ belongs to the space W^(0; R N ) if (0.3) u e Htf(il;R N ) i V'- 2 («)l«2 S LU^R); V«- 2 (u)»«2 6 ^(0; JS) We say that u G 7il^l(Q f lr N ) is a weak solution of (0,4) diva(x,u(x) y Du(x))= 0 a: G ft if (0.5) / (a(ar, u(x), Du(x))\D p(x))dx = 0 Vy?: ft -+ III with supp p CC ft, verifying V G /^(ftjir ), V q -\u)\v? G ^oc (Q;IR), K»-» P^ 2 G 1^(0; IR). We recall that L i,s (Q;R) g GL^ftjIR) such that is the Morrey space, that is the set of all sup(r~ s / \g{x)\ l dx : y G ft ; 0 r diarno) -foo JflC\B(y,r) 42 where B(y, r) is the open ball of radious r, around y. We say that g e -L\i*.(Q; HI) if g e L its (A\JR,) for every bounded open set A : A CC 0-. With regard to the function / which appears in (0.2), we suppose that (0.6) fel&(q;m.) where the exponents t,s are given by the following relations: s = Xt/2 for some A e (0, n]; if n = 2 we assume * 2; if n 3 we assume * = 2qn/{qn (q 2)(n 2)}. Now we are able to state our regularity result: THEOREM. Let CI be an open subset of R n,n 2; for q 2, let u e n^gc(n;hl N ), N 1, verify the equation (0.5). We suppose that (0.1), (0.2), (0.6) hold; then there exists // 0 such that (1.1) V-*(u)\Du\'e LgWiVt.) A straightforward consequence of the previous theorem is the following COROLLARY In the situation of the previous theorem, if n = 2, then there exists a 0 such that u C^(Q; IR N ). We'll prove our theorem by means of the technique used by K.O.Widrnan in [9]. ~~ Proof of the theorem. Let Qi be a bounded open set such that Qi CC ft; there exists another bounded open set Q2 such that Qi CC ^2 CC ft. Let us set d = dist(qi\dq2)\ we fix y fti and r (0,.cJ]; let B r = B(y,r) be the open ball of radious r, around y; let 77 be a function verifying 77 e C^(B(y,r);IR)., 0 77 1, 77 = 1 on B(y,r/2), \Di]\ 4/r. Let f belong to IR N : f will be chosen later. We insert ip = rj 2 (u - ) in (0.5): (1.2) (/)= / (a(x,u,du)\n 2 Du)dx = - [ (a(x,u,du)\2rj(u- QDrfidx = (II). By means of (0.1) we get (1.3) v f V q - 2 (u)\du\ 2 r) 2 dx (I) JB r 43 ' Now we use (0.2) and inequality 2AB A 2 + B 2 : (1.4) (II) im l V«~ 2 (u)\du\ 2 Ti 2 dx + em f V*- 2 {u)fwdx+ 4- (2M/c) / V q - 2 (u)\drj\ 2 \u - \ 2 dx Vf 0. J B r r We select c = v/(2m) and we insert (1.3), (1.4) in (1.2): (1.5) / V q - 2 (u)\du\wdx 8(M/i/) 2 7^ / V^MlD^lu-^daH- ^ -f / V q - 2 (u)fri l dx. JB r Now we use the following inequality (see [2,p. 151]): for every q 2, for every integer N 1 there exists ci 0 such that (1.6) V*- 2 (ti,) u; - v\ 2 c x \V^- 2^2(w)w - V. r 2 V 2 (vh 2 Vu;,vG Hi . We apply (1.6) and the properties of r/: / V q ~ 2 (v)\drtf\u-tfdx c 2 r- 7! / \V^- 2 V 2 (u)u- (1.7) JBr JBr-Br/* - V^- 2^2(Of J 2 * = (J//). : Now we choose in this way: is the unique vector G IR^ such that V(*-2)/2 (^ = { meas [B r - A/d} 1 / V ( *- 2)/2 (u)t/ fz JB r -B r /2 We note that V(«- 2 V 2 (ti)t* e // 1-2 (B r ;IR iv ) and D[K^~ 2 )/ 2 (u)w] 2 (g/2) 2 K 9_2 (u) Diz 2 so, by means of Poincare inequality (1.8) ('///) c 3 / V r «- 2 (ti) D«2 da;. JBr-Br/2 We use hypothesis (0.6), Sobolev imbedding theorem and Holder inequality (1-9) / V q - 2 (u)f 2 rj 2 dx c A r x. 44 We recall that»/ = lon B r j 2 and we insert (1.7), (1«8), (1.9) in (1.5): / V q ~ 2 (u)\du\ 2 dx c 5 / V q ~ 2 (u)\du\ 2 dx + c 4 i We fill the hole , that is, we add c 5 / Br/2 V q ~ 2 {u)\du\ 2 dx of the previous inequality; we get to both sides / V q ~ 2 (n)\du\ 2 dx {c B /(l.+ C 5 )}( / V q ~ 2 (u)\du\ 2 dx + (c 4 /c 5 )r x ). JB r/2 \JB r J We set 0 = max{2~ A / 2 ;c 5 /(l + C5)}. We summarize all the previous calculations into the following formula: there exists 6 E (2~ A, 1) such that (1.10) / V q - 2 {u)\du\ 2 dx 0\ f V q - 2 (u)\du\ 2 dx + (c 4 /c 5 )r x ) JB{y y r/2) \^(y,r) / for every y -,fii, for every r 6-(0,rf. By iteration we get, for every integer k 1: (1.11) / V q - 2 (u)\du\ 2 dx e k ( f V q - 2 (u)\du\ 2 dx + {c A /c* )r x V]{e2 x )' i \ JB(y,r/2k) V/jB(y,r) ^J / If /? G (0,r], then r/2 k+1 p r/2 k for some integer k; we note that x 1; we set -ji = (-.log 0)/log 2, c 6 = (c 4 /c 5 jy J (^X)~ i and we apply (1.11): (1.12) / V q - 2 {u)\du\ 2 dx (p/ryo- l ( f V q ~ 2 (u)\du\ 2 dx + r x c 6 )»=o for every y e fti, for every p,r: 0 p r d. Inequality (1.12) guarantees that V q ~ 2 {u)\du\ 2 e L^^Q^R); since tit CC ft, we get (1.1). ' ; D Proof of the Corollary. By means of our theorem, since V(u) 1, we get Du e ^(ftijir ^). We can apply a result due to Morrey [6. p. 79], [2, p. 64] and we get our thesis. ( 45 REFERENCES [1] Campanato S., Maggiorazioni fondamentali all'interno per certe classi di operatori ellittici del II ordine, Lecture given at the International meeting in memory of G.Stampacchia, Roma [2] Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton University Press, Princeton [3] Giusti E., Regolarita parziale delle soluzioni di sistemi ellittici quasi lineari di ordine arbitrario, Ann. ScuolaNorm. Sup. Pisa CI. Sci 23 (3) (1969), [4] Giusti E., Un'aggiunta alia mia nota: Regolarita parziale delle soluzioni di sistemi ellittici quasi lineari di ordine arbitrario, Ann. Scuola Norm. Sup. Pisa CI. Sci. 27 (3) (1973), [5] Leonetti F., Regolarita parziale delle soluzioni di una classe di sistemi ellittici non lineari, Matematiche (Catania) 36 (1981), [6] Morrey C.B. Jr., Multiple integrals in the calculus of variations, Springer Verlag, New York [7] Morrey C.B. Jr., Partial regularity results for nonlinear elliptic systems, Journal of Mathematics and Mechanics 17 (1968), [8] Nirenberg L., Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), [9] Widman K.O., Holder continuity of solutions of elliptic systems, Manuscripta Math. 5 (1971), Francesco LEONETTI, Dipartimento di Matematica Pura ed Applicata Universita di L'Aquila, L'Aquila, ITALY Lavoro pervenuto in redazione il
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