Higher Integrability for Solutions to Nonhomogeneous Elliptic Systems in Divergence Form

Transcription

1 Mathematica Aeterna, Vol. 4, 2014, no. 3, Higher Integrability for Solutions to Nonhomogeneous Elliptic Systems in Divergence Form GAO Hongya College of Mathematics and Computer Science, Hebei University, Baoding, , China MA Dongna College of Mathematics and Computer Science, Hebei University, Baoding, , China LI Shuangli College of Mathematics and Computer Science, Hebei University, Baoding, , China Abstract In this paper we obtain a higher integrability result for weak solutions to nonhomogeneous elliptic systems in divergence form under some suitable assumptions. Mathematics Subject Classification: 35J62, 35J47, 35D10. Keywords: Higher integrability, nonhomogeneous elliptic system, weak Lebesgue space. 1 Introduction and Statement of Result Let R n, n 3, be a bounded domain and N 2 be an integer. For i,j 1,2,,n and α,β 1,2,,N, we let the Carathéodory functions a αβ ij (x,y) : R N R to be bounded, i.e., there exists a constant c 1 such that a αβ ij (x,y) c 1, (1.1) for almost every x, for every y R N, for all i,j 1,2,,n, and for all α,β 1,2,,N.

2 258 GAO Hongya, MA Dongna, LI Shuangli We consider weak solutions u (u 1,,u N ) W 1,2 (,R N ) to nonhomogeneous elliptic systems in divergence form N D i a αβ ij (x,u(x))d j u β (x) D i fi α (x,u(x)), x, α 1,,N. j1β1 (1.2) Definition 1.1 A function u W 1,2 (,R N ) is called a weak solution to (1.2), if N α,β1 a αβ ij (x,u(x))d j u β (x)d i v α (x)dx holds true for any v W 1,2 0 (,R N ). N fi α (x,u(x))d iv α (x)dx α1 (1.3) In [1] the authors consider homogeneous elliptic system N D i a αβ ij (x,u(x))d j u β (x) 0, x, α 1,,N, j1β1 and assume, besides the ellipticity condition of diagonal coefficients, that offdiagonal coefficients a γβ ij (x,y), i,j 1,2,,n, γ,β 1,2,,N, γ β, are small when the corresponding component y γ is large, i.e., there exists q,c 2 > 0 such that 0 < θ γ u γ a γβ ij (x,y) c 2 for β γ. (1.4) y γ q The authors obtained an estimate for the measure of the superlevel set: for every s > 0, { u γ > s} c 3 s 2 (1+q), where c 3 isaconstant. Thatis, u γ isintheweak Lebesguespace withexponent 2 (1+q), u γ L 2 (1+q) weak (). See Theorem 2.2 in[1]. For other related works, see [2-7]. Inthe present paper, we deal with integrability property for weak solutions u to the nonhomogeneous elliptic system (1.1). We assume ellipticity of diagonal coefficients ij (x,y) for large values of the corresponding component of y: for γ 1,2,,N, there exists θ γ > 0 such that θ γ y γ ν ξ 2 ij (x,y)ξ i ξ j (1.5) for almost every x and for any ξ (ξ 1,ξ 2,,ξ n ) R n, where ν > 0 is a constant. For off-diagonal coefficients a γβ ij (x,y), i,j 1,2,,n, γ,β

3 Higher Integrability for Solutions to Nonhomogeneous Elliptic Systems 259 1,2,,N, γ β, weassume (1.4). Forthetermsf γ i (x,u(x)), γ 1,2,,N, we assume that they are small when the corresponding component y γ is large: 0 < θ γ u γ f γ i (x,y) c 2 y γ q, γ 1,2,,N. (1.6) The main result of this paper is Theorem 1.2 Suppose that for every γ 1,2,,N, < inf uγ and supu γ < +. (1.7) Under the previousassumptions(1.1), (1.4), (1.5)and (1.6), letu (u 1,u 2,,u N ) be a weak solution of the system (1.2), then u attains higher integrability u L 2 (1+q) weak (,R N ). 2 Proof of Theorem 1.1. For every γ {1,2,,N}, for every L (0,+ ), we define g1(l) γ max max sup sup a γβ ij (x,y) (2.1) i,j β γ x and g γ 2(L) max i y γ >L sup y γ >L sup f γ i (x,y). (2.2) x By the conditions (1.1), (1.4) and (1.6) we have { 0 < g1(l),g γ 2(L) γ min c 1, c } 2. (2.3) L q Our nearest goal is to prove that for every γ 1,2,,N, the estimate {x : u γ (x)} > 2L holds true for L max{θ γ,sup u γ, inf u γ }, where c 4 2(n 1) n 2 n 2 (N 1) Du L 2 () ν ( g c 1(L) γ 4 L +c g γ ) 2 2(L) 5, (2.4) L, c 5 2(n 1) n 1 2, n 2 ν and u γ is the γ-th component of u (u 1,,u N ). Since L sup u γ, we have (u γ L) + W 1,2 0 (). We define v (v 1,v 2,,v N ) as follows { v α 0, if α γ, v γ (u γ L) + (2.5), otherwise.

4 260 GAO Hongya, MA Dongna, LI Shuangli This implies { Dv α 0, if α γ, Dv γ 1 {u γ Du γ, otherwise, (2.6) where 1 E is the characteristic function of the set E. We insert such a test function v into (1.3), Then f γ {u γ i (x,u(x))d i u γ (x)dx N fi α (x,u(x))d iv α (x)dx α1 N a αβ ij (x,u(x))d j u β (x)d i v α (x)dx α,β1 N a γβ ij (x,u(x))d j u β (x {u γ (x)d i u γ (x)dx β1 {u γ ij (x,u(x))d j u γ (x)d i u γ (x)dx + a γβ {u γ ij (x,u(x))d j u β (x)d i u γ (x)dx. β γ ij (x,u(x))d j u γ (x)d i u γ (x)dx {u γ {u γ β γ + f γ {u γ i (x,u(x))d i u γ (x)dx. a γβ ij (x,u(x))d j u β (x)d i u γ (x)dx Since L θ γ, we can use ellipticity (1.5) and we get {u γ (2.7) ij (x,u(x))d j u γ (x)d i u γ (x)dx ν Du γ 2 dx. (2.8) {u γ We keep in mind the definitions (2.1) for g1(l) γ and (2.2) for g2(l) γ and we derive a γβ {u γ ij (x,u(x))d j u β (x)d i u γ (x)dx β γ (2.9) n 2 (N 1)g1(L) γ Du Du γ dx {u γ and {u γ f γ i (x,u(x))d i u γ (x)dx ng2(l) γ Du γ dx. (2.10) {u γ

5 Higher Integrability for Solutions to Nonhomogeneous Elliptic Systems 261 Equality (2.7) and estimates (2.8), (2.9), (2.10) merge into ν Du γ 2 dx n 2 (N 1)g1(L) γ Du Du γ dx+ng2(l) γ {u γ {u γ {u γ Du γ dx. (2.11) We use Hölder s inequality on the right hand side of (2.11) in order to get ν Du γ 2 dx {u γ n 2 (N 1)g γ 1(L) ( +ng γ 2(L) {u γ > L} 1 2 {u γ ( Du 2 dx 2 ( Du γ 2 dx {u γ {u γ 2. Du γ 2 dx We divide both sides by ( {u γ Duγ 2 dx 2 and using the fact {u γ > L} we get ( Du γ 2 2 n 2 (N 1)g γ ( dx 1(L) Du 2 2 ng dx + 2(L) γ 1 2. {u γ ν {u γ ν (2.12) We keep in mind that v γ (u γ L) + W 1,2 0 () and n 3, thus Sobolev inequality and (2.12) allow us to write (u γ L) 2 dx {u γ ) 2 ( 2(n 1) v γ 2 L 2 () n 2 Dvγ L 2 () 2(n 1) ( 2 Du γ 2 2 dx n 2 {u γ 2(n 1) n2 (N 1)g1(L) Du γ L 2 () n 2 ν (c 4 g γ 1(L)+c 5 g γ 2(L)) 2. + ngγ 2(L) 1 2 ν 2 2 (2.13) Since L > 0, it turns out that {u γ > 2L} {u γ > L}, thus L 2 {u γ > 2L} (2L L) 2 dx {u γ >2L} (u γ L) 2 dx (u γ L) 2 dx. {u γ >2L} {u γ (2.14) Inequality (2.13) and (2.14) merge into { u γ (x) > 2L} ( g c 1(L) γ 4 L +c g γ ) 2 2(L) 5. (2.15) L

6 262 GAO Hongya, MA Dongna, LI Shuangli ThisestimateholdstrueforeveryL max{θ γ ;sup u γ } > 0. Since inf u γ sup ( u γ ), if L max{θ γ ; inf u γ } > 0, then we can apply the previous inequality (2.15) to u. This proves (2.4). (2.4) together with (2.3) and the fact < gives us { u γ > 2L} c 6 L 2 (1+q), where c 6 is a constant. This ends the proof of Theorem 1.1. References [1] Leonetti F., Petricca P.V., Integrability for solutions to quasilinear elliptic systems, Comment. Math. Univ. Carolin., 2010, 51(3), [2] De Giorgi E., Un esempio di estremali discontimue per un problema variazionale di tipo ekkuttuco, Boll. Un. Mat. Ital., 1968, 3, [3] Nečas J., Stará J., Principio di massimo per i sistemi ellittici quasi-lineari non diagonali, Boll. Un. Mat. Ital., 1972, 6, [4] Mandras F., Principio di massimo per una classe di sistemi ellittici degeneri quasi lineari, Rend. Sem. Fac. Sci. Univ. Cagliari 1976, 46, [5] Leonardi S., A maximum principle for linear elliptic systems with discontinuous coefficients, Comment. Math. Univ. Carolin., 2004, 45, [6] Giusti E., Direct methods in the calculus of variations, World Scientific, River Edge, NJ, [7] Šverák V., Yan X., Non-Lipschitz minimizers of smooth uniformly convex functionals, Proc. Natl. Acad. Sci. USA, 2002, 99, Received: March, 2014