REGULARITY FOR NON-UNIFORMLY ELLIPTIC SYSTEMS AND APPLICATION TO SOME VARIATIONAL INTEGRALS. Francesco Leonetti and Chiara Musciano

Transcription

1 Electronic Journal of Differential Equations Vol. 1995(1995), No. 6, pp Published June 7, ISSN URL: or ftp (login: ftp) or REGULARITY FOR NON-UNIFORMLY ELLIPTIC SYSTEMS AND APPLICATION TO SOME VARIATIONAL INTEGRALS Francesco Leonetti and Chiara Musciano Abstract. This paper deals with higher integrability for minimizers of some variational integrals whose Euler equation is elliptic but not uniformly elliptic. This setting is also referred to as elliptic equations with p, q-growth conditions, following Marcellini. Higher integrability of minimizers implies the existence of second derivatives. This improves on a result by Acerbi and Fusco concerning the estimate of the (possibly) singular set of minimizers.. Introduction Let Ω be a bounded open set of R n, n, u be a (possibly) vector-valued function, u :Ω R N,N 1, F be a continuous function, F : R nn R; we consider the integral I(u) = F(Du(x)) dx, (.1) Ω where F (ξ) c(1 + ξ p ), (.) u W 1,p (Ω), p. Regularity of minimizers has been widely studied when ˆm (1 + ξ p ) λ DDF(ξ)λλ, < ˆm, (.3) DDF(ξ) c(1 + ξ p ), (.4) see [4], [14], [16], [17] (and [1], [1], [18], [11], [], where (.3) is weakened in order to consider quasi-convex integrals but (.4) is still present). We refer to (.3), (.4) as uniform ellipticity condition. When dealing with Ĵ(u) = {a Du + a Du p + 1+(detDu) } dx. (.5) Ω where n p<n,u:r n R n,a>, it turns out that (.4) does not hold true any longer; conversely, the following growth condition applies: DDF(ξ) c(1 + ξ n ). (.6) 1991 Mathematics Subject Classifications: 35J6, 49N6. Key words and phrases: Regularity, weak solutions, minimizers, ellipticity, variational integrals. c 1995 Southwest Texas State University and University of North Texas. Submitted: September, This work has been supported by MURST, GNAFA-CNR and INdAM. 1

2 F. Leonetti & C. Musciano EJDE 1995/6 Moreover, if a is large enough [13], namely a a(n) =n 4 [(n )!], then (.3) is still true: we are lead to consider integrals (.1) verifying (.), (.3) and DDF(ξ) c(1 + ξ q ), (.7) for some q>p. We refer to (.3), (.7) as nonuniform ellipticity condition [13], nonstandard growth condition [], or p, q-growth condition [3]. In this paper we prove higher integrability and differentiability for minimizers of integrals verifying the nonuniform ellipticity (.3), (.7). Our results apply to the model integral (.5) in this way: assume that u : R n R n, u W 1,p (Ω), Ω R n is bounded and open, n n <p<n;ifuminimizes Ĵ and a a(n), then DDu and D( Du (p )/ Du) L loc(ω). (.8) We can also apply a partial regularity theorem contained in [1], see also [15], in order to get Du C,α loc (Ω ), α (, 1), (.9) for some open Ω Ω, with Ω \ Ω =, (.1) where E is the n-dimensional Lebesgue measure of E R n. Nowweareableto improve on (.1), because of our result (.8): H n +ɛ (Ω \ Ω )= ɛ>, (.11) where H n +ɛ is the (n +ɛ)-dimensional Hausdorff measure. 1. Notation and main results Let Ω be a bounded open set of R n, n, u be a (possibly) vector-valued function, u :Ω R N,N 1, F be a function F : R nn R. We consider the integral I(u) = F(Du(x)) dx, (1.1) Ω where F C 1 (R nn ) (1.) and, for some positive constants c, p, m, F (ξ) c(1 + ξ p ), (1.3) DF(ξ) c(1 + ξ p 1 ), (1.4) m ( ξ + ˆξ ) (p ) ξ ˆξ (DF(ξ) DF(ˆξ))(ξ ˆξ), (1.5) for every ξ, ˆξ R nn.aboutp, we assume that p. (1.6) We say that u minimizes the integral (1.1) if u :Ω R N,u W 1,p (Ω) and I(u) I(u + φ). (1.7) for every φ :Ω R N with φ W 1,p (Ω). We will prove the following higher integrability result for Du:

3 EJDE 1995/6 Non-uniformly elliptic systems. 3 Theorem 1. Let u W 1,p (Ω) minimize the integral (1.1) and F satisfy (1. 1.6); then Du L σ loc(ω), σ <p n n 1. (1.8) The higher integrability result (1.8) for Du allows us to get existence of second weak derivatives under additional conditions on F. Now we assume that and, for some constants c, p, q, ˆm, µ, F C (R nn ) (1.9) F (ξ) c(1 + ξ p ), (1.1) DF(ξ) c(1 + ξ p 1 ), (1.11) ˆm (µ + ξ p ) λ DDF(ξ)λλ, < ˆm, µ, (1.1) DDF(ξ) c(1 + ξ q ), (1.13) p<q<p n n 1, (1.14) for every ξ,λ R nn. Let us remark that (1.1) implies (1.5): compare with Corollary.8 in the next section. Theorem. Let u W 1,p (Ω) minimize the integral (1.1) and F satisfy ( ); then D( Du (p )/ Du) L loc(ω). (1.15) Moreover, if (1.1) holds true with <µ,then DDu L loc(ω). (1.16) In this setting we can apply the partial regularity result contained in [1], see also [15],inordertoget Du C,α loc (Ω ), α (, 1), (1.17) for some open Ω Ω, with Ω \ Ω =. (1.18) Now Theorem allows us to improve on the estimate (1.18) of the (possibly) singular set. This is achieved in the following: Theorem 3. Let u W 1,p (Ω) minimize the integral (1.1) and F satisfy ( ); moreover, we assume that (1.1) holds true with <µ: then, there exists an open set Ω, Ω Ω, such that Du C,α loc (Ω ), α (, 1) (1.19) and H n +ɛ (Ω \ Ω )=, ɛ>, (1.) where H n +ɛ is the (n +ɛ)-dimensional Hausdorff measure.

4 4 F. Leonetti & C. Musciano EJDE 1995/6 A model functional for the previous theorems is J(u) = {a Du + a Du p + h(det Du)} dx, (1.1) Ω where u : R n R n, h : R R, h C (R), and for some positive constants c 1,d, 1 d<, (1.) h(t) c 1 (1 + t ) d, (1.3) h (t) c 1 (1 + t ) d 1, (1.4) h (t) c 1, (1.5) for every t R. Under these assumptions if a is large enough, see [13], and a a(n, d, c 1 )=c 1 n 4 [(n )!]{1+[n!] d 1 } (1.6) n n <p<n, nd p, (1.7) then (1.9),..., (1.14) hold true with q =nand µ = 1 in (1.1). For example, we can take h(t) = 1+t,d=1,c 1 = 1; the resulting functional is Ĵ(u) = {a Du + a Du p + 1+(detDu) } dx. (1.8) Ω In order to deal with weak solutions of non-uniformly elliptic systems which are not Euler equations of variational integrals, we find out that Theorem 1 remains true; with regard to Theorem, we need a more restrictive range of q. More precisely, we consider A : R nn R nn and the system of partial differential equations where and for some positive constants c, p, m, div ( A(Du(x)) ) =, (1.9) A C (R nn ), (1.3) A(ξ) c(1 + ξ p 1 ), (1.31) m ( ξ + ˆξ ) (p ) ξ ˆξ (A(ξ) A(ˆξ))(ξ ˆξ), (1.3) for every ξ, ˆξ R nn.aboutp, we keep on assuming p. (1.33) We say that u is a weak solution of (1.9) if u :Ω R N,u W 1,p (Ω) and A(Du(x)) Dφ(x) dx =, (1.34) Ω for every φ :Ω R N with φ W 1,p (Ω). We have the following higher integrability result for Du:

5 EJDE 1995/6 Non-uniformly elliptic systems. 5 Theorem 4. Let u W 1,p (Ω) be a weak solution of (1.9) and A satisfy ( ); then Du L σ loc(ω), σ <p n n 1. (1.35) As in the case of minimizers, the higher integrability of Du allows us to get higher differentiability; let us remark that, when dealing with elliptic systems that are not the Euler equation of some variational integral, we do not know any longer that the bilinear form (λ, ξ) DA λ ξ is symmetric: this lack of information is responsible for the more restrictive range of q in the following (1.4). Now we assume that and, for some constants c, p, q, ˆm, µ, A C 1 (R nn ), (1.36) A(ξ) c(1 + ξ p 1 ), (1.37) ˆm (µ + ξ p ) λ DA(ξ)λλ, < ˆm, µ, (1.38) DA(ξ) c(µ+ ξ q ), (1.39) p<q<p n 1 n, (1.4) for every ξ,λ R nn. Let us remark that (1.38) implies (1.3). Theorem 5. Let u W 1,p (Ω) be a weak solution of (1.9) and A satisfy ( ); then D( Du (p )/ Du) L loc(ω). (1.41) Moreover, if (1.38), (1.39) hold true with <µ,then DDu L loc(ω). (1.4) Remark. The most important result of this paper is Theorem 1: in our framework, the main step towards regularity is the improvement from Du L p to Du L σ, σ<pn/(n 1), which is contained in Theorem 1. This higher integrability result is achieved by a careful use of difference quotient technique: when dealing with DF(Du(x + he s )) DF(Du(x)), where h R and e s is the unit vector in the x s direction, we do not use any second derivatives of F but we employ growth condition (1.4) for DF: DF(ξ) c(1 + ξ p 1 ); this allows us to gain only a fractional derivative of Du (p )/ Du but it is enough in order to improve on the integrability of Du: see (3.5), (3.6) with the discussion between (4.1) and (4.5). This proof collects some ideas found in [7], [6], [9], [7], [5]. Remark. Regularity for scalar minimizers of variational integrals and scalar weak solutions to elliptic equations with p, q-growth condition (.3), (.7) can be found in [], [3].

6 6 F. Leonetti & C. Musciano EJDE 1995/6. Preliminaries For a vector-valued function f(x), define the difference τ s,h f(x) =f(x+he s ) f(x), where h R, e s is the unit vector in the x s direction, and s =1,,...,n. For x R n,let (x ) be the ball centered at x with radius R. We will often suppress x whenever there is no danger of confusion. We now state several lemmas that are crucial to our work. In the following f :Ω R k,k 1; B ρ,, B ρ and B R are concentric balls. Lemma.1. If <ρ<r, h <R ρ,1 t<,s {1,...,n}, f, D s f L t ( ),then τ s,h f(x) t dx h t D s f(x) t dx. B ρ (See [14,p. 45], [5,p. 8].) Lemma.. Let f L t (B ρ ), 1 <t<,s {1,...,n}; if there exists a positive constant C such that τ s,h f(x) t dx C h t, B ρ for every h with h <ρ, then there exists D s f L t (B ρ ). (See [14,p. 45], [5,p. 6].) Lemma.3. If f L (B 3ρ ) and for some d (, 1) and C> n s=1 B ρ τ s,h f(x) dx C h d, for every h with h <ρ,thenf L r (B ρ/4 ) for every r<n/(n d). Proof. The previous inequality tells us that f W b, (B ρ/ ) for every b<d,sowe can apply the embedding theorem for fractional Sobolev spaces. [3,chapter VII]. Lemma.4. For every t with 1 t<, for every f L t (B R ), for every h with h <R, for every s =1,,...,n we have f(x + he s ) t dx f(x) t dx. B R Lemma.5. For every p ) τ s,h ( f(x) (p )/ ( f(x) k 3 p ) 1 f(x)+tτ s,h f(x) p τ s,h f(x) dt for every f L p (B R ), for every h with h <R, for every s =1,,...,n,for every x.

7 EJDE 1995/6 Non-uniformly elliptic systems. 7 Lemma.6. For every γ> 1, for every k N there exist positive constants c,c 3 such that c ( v + w ) γ/ 1 v + tw γ dt c 3 ( v + w ) γ/ (.1) for every v, w R k. (See [ ].) Lemma.6 allows us to easily get the following Corollaries. Corollary.7. For every p, for every k N there exists a positive constant c 4 such that 1 c 4 λ + t(ξ λ) p dt ( λ + ξ ) p (.) for every λ, ξ R k. Corollary.8. Let F be a function F : R nn R of class C (R nn ) and p ; if there exists ˆm >such that ˆm ξ p λ DDF(ξ)λλ, for every ξ,λ R nn, then there exists m>such that m ( ξ + ˆξ ) (p ) ξ ˆξ (DF(ξ) DF(ˆξ))(ξ ˆξ), for every ξ, ˆξ R nn. Corollary.9. For every p, for every k N there exists a positive constant ĉ such that λ ξ p ĉ λ p p λ ξ ξ (.3) for every λ, ξ R k. 3. Proof of Theorem 1 Since u minimizes the integral (1.1) with growth conditions as in (1.3), (1.4), u solves the Euler equation, DF(Du(x))Dφ(x) dx =, (3.1) Ω for all functions φ :Ω R N,withφ W 1,p (Ω). Let R> be such that B 4R Ω and let B ρ and be concentric balls, <ρ<r. Let η : R n R be a cut off function in C ( )withη 1onB ρ, η 1. Fix s {1,...,n},take < h <R.Usingφ=τ s, h (η τ s,h u) in (3.1) we get, as usual (I) = η τ s,h (DF(Du)) τ s,h Du dx = τ s,h (DF(Du)) ηdη τ s,h udx=(ii) (3.)

8 8 F. Leonetti & C. Musciano EJDE 1995/6 We apply (1.5) so that m ( Du(x + he s ) + Du(x) ) p τ s,h Du(x) η (x) dx (I). (3.3) Now we use Lemma.5 and Corollary.7 in order to get, for some positive constant c 5, independent of h, c 5 τ s,h ( Du(x) Du(x)) (p )/ η (x) dx m ( Du(x + he s ) + Du(x) ) p τ s,h Du(x) η (x) dx. (3.4) In order to estimate (II), we first use the growth condition (1.4): τ s,h (DF(Du(x))) = DF(Du(x + he s )) DF(Du(x)) DF(Du(x + he s )) + DF(Du(x)) c (1 + Du(x + he s ) p 1 )+c(1 + Du(x) p 1 ). (3.5) We apply inequality (3.5) and the properties of the cut off function η, thenhölder inequality, finally Lemma.1 and.4: (II) c 6 (1 + Du(x + he s ) p 1 + Du(x) p 1 ) τ s,h u(x) dx (p 1)/p 1/p c 7 (1 + Du(x + he s ) p + Du(x) p ) dx τ s,h u(x) p dx (p 1)/p 1/p c 8 (1 + Du(x) p ) dx D s u(x) p dx h c 9 h, (3.6) B R for some positive constants c 6,c 7,c 8,c 9 independent of h. Collecting the estimates for (I) and(ii) yields, for some positive constant c 1, independent of h, B R τ s,h ( Du(x) (p )/ Du(x)) η (x) dx c 1 h, (3.7) for every s =1,...,n, for every h with h <R.Sinceη=1onB ρ, inequality (3.7) allows us to apply Lemma.3 in order to get Du (p )/ Du L r (B ρ/4 ), r <n/(n 1). We remark that Du (p )/ Du = Du p/, thus (1.8) is completely proven.

9 EJDE 1995/6 Non-uniformly elliptic systems Proof of Theorem We start as in the proof of Theorem 1 and we arrive at (3.); now F has second derivatives, thus τ s,h (DF(Du(x))) =DF(Du(x + he s )) DF(Du(x)) 1 d ( = DF(Du(x)+tτs,h Du(x)) ) dt dt = 1 DDF(Du(x)+tτ s,h Du(x))τ s,h Du(x) dt. (4.1) Let us remark that second derivatives of F verify (1.13) with p<q: (4.1) is not very useful when carrying on the standard difference quotient technique if we only know that Du L p. But we have already proven, in Theorem 1, that Du L σ, for every σ<pn/(n 1). Since we assumed (1.14), then Du L q and we can go on using (4.1) in (3.): 1 DDF(Du + tτ s,h Du) ητ s,h Du η τ s,h Du dt dx =(I) =(II)= 1 DDF(Du + tτ s,h Du) ητ s,h Du Dη τ s,h udtdx. (4.) Since F is C, the bilinear form (λ, ξ) DDF(Du + tτ s,h Du) λξ is symmetric; moreover, it is positive because of (1.1). Therefore we can use Cauchy-Schwartz inequality in order to get (II) DDF(Du + tτ s,h Du) ητ s,h Du η τ s,h Du dt dx DDF(Du + tτ s,h Du) Dη τ s,h udητ s,h udtdx (4.3) = 1 (I)+(III). As we have already pointed out, in Theorem 1 we have proven the higher integrability (1.8), so that, with the aid of (1.14), we can get Du L q loc (Ω). (4.4) Growth condition (1.13) and higher integrability (4.4) make the two integrals in (4.3) finite, so we can subtract 1 (I) from both sides of (4.) in order to get 1 (I) (III). (4.5)

10 1 F. Leonetti & C. Musciano EJDE 1995/6 Let us estimate (III). First we use the properties of the cut-off function η with the growth condition (1.13), then Hölder inequality, finally Lemma.1 and.4 (Lemma.1 is avaliable with t = q because of (4.4) ): (III) c 11 (1 + Du(x) + Du(x + he s ) ) q τ s,h u dx q q q c 1 (1 + Du(x) q + Du(x + he s ) q ) dx τ s,h u q dx c 13 (1 + Du q ) dx h = c 14 h, B R (4.6) for some positive constants c 11,c 1,c 13,c 14 independent of h. Now we estimate (I) from below: using (1.1) we have, for some positive constant c 15 independent of h, µc 15 τ s,h Du η dx + c 15 τ s,h ( Du (p )/ Du) η dx (I). (4.7) Collecting the previous inequalities yields, for some positive constant c 16 independent of h, µ τ s,h Du η dx + τ s,h ( Du (p )/ Du) η dx c 16 h, (4.8) for every s =1,...,n, for every h with h <R. Since η =1onB ρ, inequality (4.8) allows us to apply Lemma. with f = Du (p )/ Du (and f = Du provided µ>), thus giving (1.15) (and (1.16), provided µ>). This ends the proof. 5. Proof of Theorem 3. We can use the partial regularity result contained in [1], see [15] too, in order to get Du C,α loc (Ω ), α (, 1), for the open set Ω defined as follows Ω = x Ω : lim(du) R nn, r lim r n r Du(y) (Du) p dy =. where (g) = B(x, r) 1 g(y) dy. So, for the singular set, we have Ω \ Ω S 1 S,

11 EJDE 1995/6 Non-uniformly elliptic systems. 11 where S 1 = { x Ω: S = } / lim(du) r or lim (Du) = r, x Ω : lim sup r n r Let us take ξ R nn such that ξ p ξ = r n Du(y) (Du) p dy p r n Du(y) (Du) p dy >. ( ) Du p Du ;then Du(y) ξ p dy =(V); we can use Corollary.9 with λ = Du(y) and, if we keep in mind the particular choice of ξ and Poincarè inequality, we get (V ) ĉ p r n Du(y) p p Du(y) ξ ξ dy =ĉ p r n c ĉ p r n Du(y) p Du(y) ( Du p Du) dy ) D ( Du(y) p Du(y) dy. Thus S x Ω : lim sup r n r ) D ( Du(y) p Du(y) dy >. Since we have proven that DDu and D( Du (p )/ Du) L loc (Ω), we can use standard technique [19], [14], in order to get (1.). proof. This ends the 6. Proof of Theorem 4 and 5 Theorem 4 is proven just in the same way as Theorem 1, so we skip it and we go to Theorem 5. Arguing as in Theorem we get 1 DA(Du + tτ s,h Du) ητ s,h Du η τ s,h Du dt dx =(I) =(II)= 1 DA(Du + tτ s,h Du) ητ s,h Du Dη τ s,h udtdx. (6.1)

12 1 F. Leonetti & C. Musciano EJDE 1995/6 Since the bilinear form (λ, ξ) DA λ ξ is no longer symmetric, we cannot use Cauchy-Schwartz inequality as we did in (4.3). Let us remark that q < p(n 1)/(n ) <pn/(n 1), so we can use the higher integrability result proven in Theorem 4: Du L σ loc(ω), σ <p n n 1. (1.35) We apply the nonuniform ellipticity conditions (1.38) and (1.39), then we use (1.35) with σ = q: ˆm 1 (µ + Du+ tτ s,h Du p ) τ s,h Du η dt dx =ˆm(IV ) (I) <. (6.) Let us estimate (II). First of all we use the growth condition (1.39): DA(Du + tτ s,h Du) ητ s,h Du Dη τ s,h u c 17 (µ + Du + tτ s,h Du q ) ητ s,h Du τ s,h u ɛ (µ + Du + tτ s,h Du p ) ητ s,h Du + c 17 ɛ (µ + Du + tτ s,hdu q p ) τ s,h u, ɛ >, for some positive constant c 17 independent of h and ɛ, sothat (6.3) (II) ɛ(iv )+ c 17 ɛ 1 (µ + Du + tτ s,h Du q p ) τ s,h u dt dx. (6.4) Because of (1.4), p<q<q p<pn/(n 1), so (1.35) allows us to use Lemma.1 and.4 with t =q pand f = Du: 1 (µ + Du + tτ s,h Du q p ) τ s,h u dt dx c 18 c 19 (µ q p q p τ s,h u q p dx + Du(x) q p + Du(x + he s ) q p) dx q p (µ q p q p + Du(x) q p ) dx h q p q p (6.5) B R c h, for some positive constants c 18,c 19,c independent of h and ɛ. Inequalities (6.4) and (6.5) give (II) ɛ(iv )+ c 1 ɛ h, (6.6)

13 EJDE 1995/6 Non-uniformly elliptic systems. 13 for some positive constant c 1 independent of h and ɛ. Now we use (6.1), (6.) and (6.6): ˆm (IV ) (I)=(II) ɛ(iv )+ c 1 ɛ h ; (6.7) we select ɛ = 1 ˆm in (6.7); since (IV ) <, we can subtract ɛ(iv )frombothsides of (6.7) thus giving ˆm (IV ) c 1 ˆm h. (6.8) This last inequality and Lemma.5 end the proof. References [1] Acerbi, E. and Fusco, N., A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Anal. 99 (1987), [] Acerbi, E. and Fusco, N., Partial regularity under anisotropic (p,q) growth conditions, J. Differential Equations 17 (1994), [3] Adams, R. A., Sobolev Spaces, Academic Press, New York, [4] Bhattacharya, T. and Leonetti, F., On improved regularity of weak solutions of some degenerate, anisotropic elliptic systems, Ann. Mat. Pura Appl. (to appear). [5] Campanato, S., Sistemi ellittici in forma divergenza. Regolarità all interno., Quaderni Scuola Normale Superiore, Pisa 198. [6] Campanato, S., Hölder continuity of the solutions of some nonlinear elliptic systems, Adv. in Math. 48 (1983), [7] Campanato, S. and Cannarsa, P., Differentiability and partial hölder continuity of the solutions of nonlinear elliptic systems of order m with quadratic growth, Ann. Scuola Norm. Sup. Pisa 8 (1981), [8] Choe, H. J., Interior behaviour of minimizers for certain functionals with nonstandard growth, Nonlinear Analysis T.M.A. 19 (199), [9] De Thelin, F., Local regularity properties for the solutions of a nonlinear partial differential equation, Nonlinear Analysis T.M.A. 6 (198), [1] Evans, L.C., Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal. 95 (1986), 7 5. [11] Evans, L.C. and Gariepy, R.F., Blow up, compactness and partial regularity in the calculus of variations, Indiana Univ. Math. J. 36 (1987), [1] Fusco, N. and Hutchinson, J., C 1,α partial regularity of functions minimizing quasiconvex integrals, Manuscripta Math. 54 (1985), [13] Frasca, M. and Ivanov, A.V., Partial regularity for quasilinear nonuniformly elliptic systems, Le Matematiche 46 (1991), [14] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 15, Princeton University Press, Princeton [15] Giaquinta, M., Quasiconvexity, growth conditions and partial regularity, in: Partial differential equations and calculus of variations, Lecture Notes in Math. 1357, Springer, Berlin, 1988, [16] Giaquinta, M. and Giusti, E., Differentiability of minima of nondifferentiable functionals, Inventiones Math. 7 (1983), [17] Giaquinta, M. and Ivert, P.A., Partial regularity for minima of variational integrals, Ark. Mat. 5 (1987), 1 9. [18] Giaquinta, M. and Modica, G., Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non linéaire 3 (1986), [19] Giusti, E., Precisazione delle funzioni H 1,p e singolarità delle soluzioni deboli di sistemi ellittici nonlineari, Boll. Un. Mat. Ital. -A (1969), [] Hong, M.C., Existence and partial regularity in the calculus of variations, Ann. Mat. Pura Appl. 149 (1987), [1] Leonetti, F., Weak differentiability for solutions to nonlinear elliptic systems with p,q-growth conditions, Ann. Mat. Pura Appl. 16 (199), [] Marcellini, P., Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal. 15 (1989),

14 14 F. Leonetti & C. Musciano EJDE 1995/6 [3] Marcellini, P., Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations 9 (1991), 1 3. [4] Morrey, C. B. Jr., Multiple integrals in the calculus of variations, Springer Verlag, New York, [5] Naumann, J., Interior integral estimates on weak solutions of certain degenerate elliptic systems, Ann. Mat. Pura Appl. 156 (199), [6] Nirenberg, L., Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), [7] Tolksdorf, P., Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl. 134 (1983), Francesco Leonetti Dipartimento di Matematica Università di L Aquila Piazzale Aldo Moro 5, 185 Roma, Italy leonetti@vxscaq.aquila.infn.it Chiara Musciano Istituto Nazionale di Alta Matematica Francesco Severi Città Universitaria Piazzale Aldo Moro 5, 185 Roma, Italy