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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 835736, 11 pages doi:10.1155/2008/835736 ResearchArticleLocalRegularityResultsforMinimaofAnisotropicFunctionalsandSolutionsofAnisotropic Equations Gao Hongya, 1, 2 Qiao Jinjing, 1 Wang Yong, 3 and Chu Yuming 4 1 College of Mathematics and Computer Science, Hebei University, Baoding 071002, China 2 Study Center of Mathematics of Hebei Province, Shijiazhuang 050016, China 3 Department of Mathematics, Chengde Teachers College for Nationalities, Chengde 067000, China 4 Faculty of Science, Huzhou Teachers College, Huzhou 313000, China Correspondence should be addressed to Gao Hongya, hongya-gao@sohu.com Received 13 July 2007; Accepted 21 November 2007 Recommended by Alberto Cabada This paper gives some localregularityresultsforminimaofanisotropicfunctionals I u; Ω Ω fx, u, Dudx, u ∈ W 1,q i loc Ω andforsolutionsofanisotropic equations −divAx, u, Du− N i1 ∂f/ ∂x i , u ∈ W 1,q i loc Ω which can be regarded as generalizations of the classical results. Copyright q 2008 Gao Hongya et al. This is an open access article distributed under the Creative Commons Attribution License, which p ermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let Ω be an open bounded subset of R N , N ≥ 2. Let q i > 1, i 1 , ,N.Denote q max 1≤i≤N q i ,p min 1≤i≤N q i , q : 1 q 1 N N i1 1 q i . 1.1 Throughout this paper, we will make use of the anisotropic Sobolev space W 1,q i loc Ω v ∈ L q loc Ω : ∂v ∂x i ∈ L q i loc Ω, ∀i 1, ,N . 1.2 Let x 0 ∈ Ω and t>0, we denote by B t the ball of radius t centered at x 0 . For functions u and k>0, let A k {x ∈ Ω : |ux| >k}, A k,t A k ∩ B t . Moreover, if p>1, then p is always the real number p/p − 1,andifs<N, s ∗ is always the real number satisfying 1/s ∗ 1/s − 1/N. This paper mainly considers the functions u minimizing the anisotropicfunctionals Iu; Ω Ω fx, u, Dudx, u ∈ W 1,q i loc Ω 1.3 2 Journal of Inequalities and Applications and weak solutionsof the anisotropic equations −divAx, u , Du− N i1 ∂f i ∂x i ,u∈ W 1,q i loc Ω. 1.4 We refer to the classical books by Lady ˇ zenskaya and Ural’ceva 1, Morrey 2, Gilbarg and Trudinger 3, and Giaquinta 4 for some details of isotropic cases. For isotropic cases, global L s -summability was proved in the 1960s by Stampacchia 5 forsolutionsof linear elliptic equations. This result was extended by Boccardo and Giachetti to the nonlinear case in 6. Foranisotropic cases, Giachetti and Porzio recently proved in 7 the local L s -summability forminimaofanisotropicfunctionalsand weak solutionsofanisotropic nonlinear elliptic equations. Precisely, the authors considered the minimaoffunctionals whose prototype is 1.3, f is a Carath ´ eodory function satisfying the growth conditions a N i1 ξ i q i ≤ fx, s, ξ ≤ b N i1 ξ i q i ϕ 1 x, 1.5 where the function ϕ 1 ∈ L r loc Ω with 1 <r<N/q. The authors also considered the localsolutions u ∈ W 1,q i loc Ω of the anisotropic equations 1.4,whereA : Ω × R × R N → R N is a Carath ´ eodory function satisfying the following structural conditions: Ax, u, ξ·ξ ≥ m 0 N i1 ξ i q i , A j x, u, ξ ≤ m 1 hx N i1 |ξ i | q i 1−1/q j ,j 1, ,N, 1.6 where m l ,l 0, 1 are positive constants, the function h is in L 1 loc Ω and the functions f i belong, respectively, to the spaces L q i loc Ω. Under the above conditions, the authors obtained some localregularity results. The aim of the present paper is to prove the localregularity property forminimaof the anisotropicfunctionalsof type 1.3 with the more general growth conditions than 1.5,that is, we assume the integrand f satisfies the following growth conditions: N i1 ξ i q i − b u α − ϕ 0 x ≤ fx, u, ξ ≤ a N i1 ξ i q i b|u| α ϕ 1 x, 1.7 where ϕ 0 ∈ L r 1 loc Ω,ϕ 1 ∈ L r 2 loc Ω,r 1 ,r 2 > 1,a≥ 1,b≥ 0, p ≤ α<p ∗ ,q<q ∗ , q<N, 1 < min r 1 ,r 2 < N q . 1.8 We also consider weak solutionsof the type 1.4 with more general growth conditions than 1.6, that is, we assume the operator A satisfies the following coercivity and growth condi- tions: Ax, u, ξ· ξ ≥ b 0 N i1 ξ i q i − b 1 |u| α 1 − ϕ 2 x, 1.9 Ax, u, ξ ≤ b 2 N i1 ξ i q i −1 b 3 |u| α 2 kx, 1.10 Gao Hongya et al. 3 where b 0 ≥ 1, b i > 0, i 1, 2, 3, q<q ∗ , q<N, p ≤ α 1 <p ∗ , p − 1 ≤ α 2 ≤ Np − 1/N − p, ϕ 2 ∈ L r 0 loc Ω with r 0 > 1, k ∈ L r N1 loc Ω, f i ∈ L r i loc Ω, i 1, ,N. Remark 1.1. Notice that we have confined ourselves to the case q<Nbecause when such inequality is violated, every function in W 1,q i loc Ω is trivially in L s loc Ω for every fixed s<∞ by 7, Lemma 3.2. Remark 1.2. Since we have assumed in 1.7, 1.9,and1.10 that the integrand f and the operator A satisfy some growth conditions depending on u, in the proof of the localregularity results, w e have to estimate the integral of some power of |u| by means of |Du|.Todothis,we will make use of the Sobolev inequality that has been used in 8. 2. Preliminary lemmas In order to prove the local L s -integrability of the local unbounded minimaof the anisotropicfunctionalsand weak solutionsofanisotropic equations, we need a useful lemma from 7. Lemma 2.1. Let u ∈ W 1,q i loc Ω, φ 0 ∈ L r loc Ω,whereq, q,andr satisfy 1 <r< N q ,q< q ∗ , q<N. 2.1 Assume that the following integral estimates hold: A k,τ N i1 ∂u ∂x i q i dx ≤ c 0 A k,t φ 0 dx t − τ −γ A k,t N i1 |u| q i dx 2.2 for every k ∈ N and R 0 ≤ τ<t≤ R 1 ,wherec 0 is a positive constant that depends only on N, q i , r, R 0 , R 1 and |Ω| and γ is a real positive constant. Then u ∈ L s loc Ω,where s q ∗ q q − q ∗ 1 − 1/r . 2.3 One will also need a lemma from [8]. Lemma 2.2. Let ft be a nonnegative bounded function defined for 0 ≤ T 0 ≤ t ≤ T 1 . Suppose that for T 0 ≤ t<s≤ T 1 , ft ≤ As − t −γ B θfs, 2.4 where A, B, γ, θ are nonnegative constants, and θ<1. Then there exists a constant c, depending only on γ and θ such that for every , R, T 0 ≤ <R≤ T 1 , one has f ≤ c AR − −γ B . 2.5 4 Journal of Inequalities and Applications 3. Minimaofanisotropicfunctionals In this section, we prove a localregularity result forminimaofanisotropic functionals. Definition 3.1. By a local minimum of the anisotropic functional I in 1.3, we mean a function u ∈ W 1,q i loc Ω, such that for every function ψ ∈ W 1,q i Ω with supp ψ ⊂⊂ Ω, it holds that Iu; supp ψ ≤ Iu ψ; supp ψ. 3.1 Theorem 3.2. Assume that the functional I satisfies the conditions 1.7.Ifu is a local minimum of I, then it belongs to L s loc Ω,where s q ∗ q q − q ∗ 1 − 1/ min r 1 ,r 2 . 3.2 Proof. Owing to Lemma 2.1,itissufficient to prove that u satisfies the integral estimates 2.2 with γ q and φ 0 ϕ 0 ϕ 1 .LetB R 1 ⊂⊂ Ω and 0 ≤ R 0 ≤ τ<t≤ R 1 be arbitrarily but fixed. It is no loss of generality to assume that R 1 − R 0 < 1. For k>0, let A k x ∈ Ω : ux >k ,A − k x ∈ Ω : ux < −k . 3.3 It is obvious that A k A k ∪ A − k .DenoteA k,t A k ∩ B t and A − k,t A − k ∩ B t .Letw maxu − k,0. Choose ψ −ηw in 3.1,whereη is a cut-off function such that supp η ⊂ B t , 0 ≤ η ≤ 1,η 1inB τ , |Dη|≤2t − τ −1 . 3.4 We obtain from the minimality of u that B t fx, u, Dudx ≤ B t fx, u ψ, Du Dψdx A k,t f x, u − ηw, Du − Dηw dx B t ∩{u≤k} fx, u, Dudx. 3.5 This implies that A k,t fx, u, Dudx ≤ A k,t f x, u − ηw, Du − Dηw dx. 3.6 By 1.7,weobtain A k,t N i1 ∂u ∂x i q i dx ≤ b A k,t u α dx A k,t ϕ 0 dx a A k,t N i1 ∂u ∂x i − ∂ηw ∂x i q i dx b A k,t u − ηw α dx A k,t ϕ 1 dx. 3.7 Gao Hongya et al. 5 We first estimate the 3rd term on the right-hand side of 3.7. Using the elementary inequality a b q ≤ 2 q−1 a q b q ,a,b≥ 0,q≥ 1, 3.8 we obtain a A k,t N i1 ∂u ∂x i − ∂ηw ∂x i q i dx a A k,t \A k,τ N i1 ∂u ∂x i − ∂ηw ∂x i q i dx ≤ 2 q−1 a A k,t \A k,τ N i1 1 − η q i ∂u ∂x i q i ∂η ∂x i q i w q i dx ≤ 2 q−1 a A k,t \A k,τ N i1 ∂u ∂x i q i dx 2 2q−1 a t − τ q i A k,t \A k,τ N i1 w q i dx ≤ 2 q−1 a A k,t \A k,τ N i1 ∂u ∂x i q i dx 2 2q−1 a t − τ q A k,t \A k,τ N i1 u q i dx 3.9 since w q i ≤ u q i in A k,t and t − τ<1. The summation of the 1st and the 4th terms on the right- hand side of 3.7 can be estimated as b A k,t u α dx b A k,t u − ηw α dx ≤ 2b A k,t u α dx. 3.10 Substituting 3.9 and 3.10 into 3.7 yields A k,t N i1 ∂u ∂x i q i dx ≤ A k,t ϕ 0 ϕ 1 dx 2b A k,t u α dx 2 q−1 a A k,t \A k,τ N i1 ∂u ∂x i q i dx 2 2q−1 a t − τ q A k,t \A k,τ N i1 u q i dx. 3.11 We know from 8 that if u ∈ W 1,p B t and | supp u|≤1/2|B t |,wethenhavetheSobolev inequality B t u p ∗ dx p/p ∗ ≤ c 1 N, p B t |Du| p dx. 3.12 Let u ⎧ ⎨ ⎩ u, x ∈ A k,t , 0,x∈ Ω \ A k,t . 3.13 6 Journal of Inequalities and Applications By assumption, p ≤ α<p ∗ , which implies A k,t u α dx B t u α dx ≤u α−p p ∗ B t 1−α/p ∗ B t u p ∗ dx p/p ∗ ≤ c 1 u α−p p ∗ B t 1−α/p ∗ B t |Du| p dx ≤ c 1 u α−p p ∗ B t 1−α/p ∗ max 1, 2 p/2−1 B t N i1 ∂u ∂x i q i dx c 1 u α−p p ∗ B t 1−α/p ∗ max 1, 2 p/2−1 A k,t N i1 ∂u ∂x i q i dx, 3.14 provided that |supp u| B t |≤1/2|B t |. We can choose T so small that for t ≤ T we get c 1 u α−p p ∗ B t 1−α/p ∗ max 1, 2 p/2−1 ≤ 1 4b . 3.15 It is obvious that k p ∗ A k ≤u p ∗ p ∗ , Ω , 3.16 and therefore, there exists a constant k 0 , such that for k ≥ k 0 ,wehave A k ≤ 1 2 B T/2 . 3.17 For such values of k we then have |supp u| < 1/2|B T/2 | and therefore, if T/2 ≤ t ≤ T, A K,t u α dx ≤ 1 4b A K,t N i1 | ∂u ∂x i | q i dx. 3.18 Thus, from 3.11 and, we get A k,t N i1 ∂u ∂x i q i dx ≤ 2 A k,t ϕ 0 ϕ 1 dx 2 q a A k,t \A k,τ N i1 ∂u ∂x i q i dx 2 2q a t − τ q A k,t \A k,τ N i1 u qi dx. 3.19 Suppose now T/2 ≤ ≤ τ<t≤ R ≤ T,weget A k, N i1 ∂u ∂x i q i dx ≤ 2 A k,R ϕ 0 ϕ 1 dx 2 q a A k,t \A k, N i1 ∂u ∂x i q i dx 2 2q a t − τ q A k,R N i1 u q i dx. 3.20 Gao Hongya et al. 7 Adding to both sides 2 q a times the left-hand side, we get eventually A k, N i1 ∂u ∂x i q i dx ≤ 2 2 q a 1 A k,R ϕ 0 ϕ 1 dx 2 q a 2 q a 1 A k,t N i1 ∂u ∂x i q i dx 2 2q a 2 q a 1 · 1 t − τ q A k,R N i1 u q i dx, 3.21 we can now apply Lemma 2.2 to conclude that A k,τ N i1 ∂u ∂x i q i dx ≤ c 2 2 q a 1 A k,t ϕ 0 ϕ 1 dx 2 2q a 2 q a 1 · 1 t − τ q A k,t N i1 |u| q i dx , 3.22 where c depends only on q and a. Since −u minimizes the functional Fv; Ω Ω fx, v, Dvdx, 3.23 where fx, v, pfx, −v, −p satisfies the same growth conditions 1.7, inequality 3.22 holds with u replaced by −u. We then conclude that A − k,τ N i1 ∂u ∂x i q i dx ≤ c 2 2 q a 1 A − k,t ϕ 0 ϕ 1 dx 2 2q a 2 q a 1 · 1 t − τ q A − k,t N i1 |u| q i dx . 3.24 Adding 3.22 and 3.24 yields A k,τ N i1 ∂u ∂x i q i dx ≤ c 2 2 q a 1 A k,t ϕ 0 ϕ 1 dx 2 2q a 2 q a 1 · 1 t − τ q A k,t N i1 |u| q i dx . 3.25 This shows that u satisfies estimates 2.2 with γ q and φ 0 ϕ 0 ϕ 1 . Theorem 3.2 follows from Lemma 2.1. 4. Localsolutionsofanisotropic equations In this section, we prove a localregularity result for weak solutionsofanisotropic equations. Let u ∈ W 1,q i loc Ω be a local solution of the anisotropic equation 1.4,whereA : Ω×R×R n → R n is a Carath ´ eodory function satisfying the structural conditions 1.9 and 1.10. Definition 4.1. By a weak solution of 1.4 we mean a function u ∈ W 1,q i loc Ω, such that for every function ψ ∈ W 1,q i Ω with supp ψ ⊂⊂ Ω it holds supp ψ Ax, u, Du·Dψ dx supp ψ f·Dψ dx, 4.1 where f f 1 ,f 2 , ,f N . 8 Journal of Inequalities and Applications Theorem 4.2. Under the previous assumptions 1.9 and 1.10, if one assumes that ϕ 2 ∈ L r 0 loc Ω, f i ∈ L r i loc Ω, i 1, 2, ,N, k ∈ L r N1 loc Ω,andr i , i 0, ,N 1 satisfy 1 <r min 1≤i≤N r i q i ,r 0 , r N1 p < N q , 4.2 then u ∈ L s loc Ω,where s q ∗ q q − q ∗ 1 − 1/r . 4.3 Proof. By virtue of Lemma 2.1,itissufficient to prove that u satisfies the integral estimates 2.2 with γ q and φ 0 ϕ 2 |k| p N i1 f i q i . Let B R 1 ⊂⊂ Ω and 0 ≤ R 0 ≤ τ<t≤ R 1 be arbitrarily but fixed. Assume again that R 1 −R 0 < 1. Let w max{u−k, 0}. Choose ψ ηw as a test function in 4.1,wherethecut-off function η satisfies the conditions 3.4.WeobtainfromDefinition 4.1 that A k,t Ax, u, Du·Dηwdx A k,t f· Dηwdx. 4.4 We now estimate the integrals in 4.4. Applying the assumption 1.9, we deduce from 4.4 that b 0 A k,τ N i1 ∂u ∂x i q i dx ≤ b 1 A k,t u α 1 dx A k,t ϕ 2 dx A k,t f·Dudx 2 t − τ A k,t f wdx 2 t − τ A k,t \A k,τ Ax, u, Du wdx. 4.5 The 3rd term on the right-hand side of the above inequality can be estimated as A k,t f· Du dx A k,t N i1 f i · ∂u ∂x i dx ≤ ε A k,t N i1 ∂u ∂x i q i dx N i1 C ε, q i A k,t f i q i dx. 4.6 By Young’s inequality, the 4th term on the right-hand side of inequality 4.5 can be estimated as 2 t − τ A k,t |f|wdx≤ 2 q t − τ q A k,t N i1 u − k q i dx N i1 A k,t f i q i dx. 4.7 By 1.10, the last term on the right-hand side of 4.5 can be estimated as 2 t − τ A k,t \A k,τ |Ax, u, Du|wdx≤ 2 t − τ A k,t \A k,τ b 2 N i1 ∂u ∂x i q i −1 b 3 |u| α 2 k wdx I 1 I 2 I 3 . 4.8 Gao Hongya et al. 9 By Young’s inequality, we derive that I 1 ≤ b 2 A k,t \A k,τ N i1 ∂u ∂x i q i dx b 2 2 q t − τ q A k,t \A k,τ N i1 u − k q i dx. 4.9 H ¨ older’s inequality and Young’s inequality yield I 2 ≤ b 3 ε A k,t \A k,τ |u| α 2 p dx Cε, p2 p t − τ p A k,t \A k,τ u − k p dx ≤ b 3 ε A k,t \A k,τ |u| α 2 p dx Cε, p2 p Nt − τ q A k,t \A k,τ N i1 u − k q i dx, 4.10 where ε is a positive constant to be determined later. Further, I 3 ≤ A k,t \A k,τ |k| p dx 2 q Nt − τ q A k,t \A k,τ N i1 u − k q i dx. 4.11 Combining 4.6–4.11 with 4.5 yields b 0 A k,τ N i1 ∂u ∂x i q i dx ≤ A k,t ϕ 2 |k| p N i1 Cε, q i 1 |f i | q i dx b 1 A k,t |u| α 1 dx b 3 ε A k,t |u| α 2 p dx ε A k,t N i1 ∂u ∂x i q i dx b 2 A k,t \A k,τ N i1 ∂u ∂x i q i dx b 2 Cε, p2 2 q t − τ q A k,t N i1 u − k q i dx. 4.12 Since p ≤ α 1 <p ∗ , then as in the proof of Theorem 3.2, we know that there exist a sufficiently small T and a sufficiently large k 0 , such that for all T/2 ≤ t ≤ T and k ≥ k 0 ,wehave A k,t |u| α 1 dx ≤ 1 2b 1 A k,t N i1 ∂u ∂x i q i dx. 4.13 Similarly, since p − 1 ≤ α 2 ≤ Np − 1/n − p,thenp ≤ α 2 p ≤ p ∗ , therefore A k,t |u| α 2 p dx ≤ C A k,t N i1 ∂u ∂x i q i dx. 4.14 10 Journal of Inequalities and Applications Thus, from 4.12–4.14 we can derive that b 0 A k,τ N i1 ∂u ∂x i q i dx ≤ A k,t ϕ 2 |k| p N i1 Cε, q i 1 f i q i dx 1 2 Cb 3 1ε A k,t N i1 ∂u ∂x i qi dx b 2 A k,t \A k,τ N i1 ∂u ∂x i q i dx b 2 Cε, p2 2 q t − τ q A k,t N i1 u − k q i dx. 4.15 Adding to both sides b 2 A k,τ N i1 ∂u ∂x i q i dx, 4.16 we get eventually A k,τ N i1 ∂u ∂x i q i dx ≤ 1 b 0 b 2 A k,t ϕ 2 |k| p N i1 C ε, q i 1 |f i | q i dx 1 2 Cb 3 1 ε 1 b 0 b 2 A k,t N i1 ∂u ∂x i q i dx b 2 b 0 b 2 A k,t N i1 ∂u ∂x i q i dx b 2 Cε, p2 1 b 0 b 2 2 q t − τ q A k,t N i1 u − k q i dx. 4.17 Choosing ε small enough, such that θ 1/2 Cb 3 1 ε b 2 b 0 b 2 < 1, 4.18 4.17 implies that A k,τ N i1 ∂u ∂x i q i dx≤C A k,t ϕ 2 |k| p N i1 |f i | q i dxθ A k,t N i1 ∂u ∂x i q i dx C t−τ q A k,t N i1 u − k q i dx. 4.19 Suppose now that T/2 ≤ ≤ τ<t≤ R ≤ T,weget A k, N i1 ∂u ∂x i q i dx ≤ C A k,t ϕ 2 |k| p N i1 |f i | q i dx θ A k,R N i1 ∂u ∂x i q i dx C R − q A k,t N i1 u − k q i dx. 4.20 [...]... e applicazioni,” Ricerche di Matematica, vol 34, pp 309–323, 1985 7 D Giachetti and M M Porzio, Localregularityresultsforminimaoffunctionalsof the calculus of variation,” Nonlinear Analysis Theory, Methods & Applications, vol 39, no 4, pp 463–482, 2000 8 M Giaquinta and E Giusti, “On the regularityof the minimaof variational integrals,” Acta Mathematica, vol 148, no 1, pp 31–46, 1982 ... 1 ϕ2 |k|p N i 1 fi qi and α N q Ak,t i 1 q Theorem 4.2 follows from Acknowledgments Gao Hongya is supported by Special Fund of Mathematics Researchof Natural Science Foundation of Hebei Province no 07M003 and Doctoral Foundation of the Department of Education of Hebei Province B2004103 Chu Yuming is supported by NSF of Zhejiang Province No Y607128 and NSFC no 10771195 This research was done when... Institute of Mathematics, Nankai University He wished to thank the Institute for support and hospitality References 1 O A Ladyˇ enskaya and N N Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New z York, NY, USA, 1968 2 C B Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, Germany, 1968 3 D Gilbarg and N S Trudinger, Elliptic Partial Differential Equations of. .. Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, no 105, Princeton University Press, Princeton, NJ, USA, 1983 ` 5 G Stampacchia, Equations Elliptiques du Second Ordre a Coefficients Discontinus, S´ minaire de e ´ Math´ matiques Sup´ rieures, Les Presses de l’Universit´ de Montr´ al, Montreal, Quebec, Canada, 1966 e e e e 6 L Boccardo and D Giachetti, “Alcune... Applying Lemma 2.2, we conclude that N Ak,τ i 1 ∂u ∂xi qi dx ≤ cC ϕ2 |k|p Ak,t N fi qi cC t−τ dx i 1 N q u−k qi dx 4.21 Ak,t i 1 Since −u is a weak solution of −div A x, u, Du − N i ∂fi , ∂xi 1 4.22 where A x, s, ξ A x, −s, −ξ satisfies the same conditions 1.9 and 1.10 , inequality 4.21 holds with u replaced by −u We then conclude that N A− i 1 k,τ ∂u ∂xi qi dx ≤ cC A− k,t ϕ2 |k|p N fi fi dx qi cC t−τ dx qi . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 835736, 11 pages doi:10.1155/2008/835736 Research Article Local Regularity Results for Minima of Anisotropic Functionals and Solutions. Cabada This paper gives some local regularity results for minima of anisotropic functionals I u; Ω Ω fx, u, Dudx, u ∈ W 1,q i loc Ω and for solutions of anisotropic equations −divAx,. in 7 the local L s -summability for minima of anisotropic functionals and weak solutions of anisotropic nonlinear elliptic equations. Precisely, the authors considered the minima of functionals