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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 878769, 12 pages doi:10.1155/2010/878769 Research Article Local Regularity and Local Boundedness Results for Very Weak Solutions of Obstacle Problems Gao Hongya, 1, 2 Qiao Jinjing, 3 and Chu Yuming 4 1 College of Mathematics and Computer Science, Hebei University, Baoding 071002, China 2 Hebei Provincial Center of Mathematics, Hebei Normal University, Shijiazhuang 050016, China 3 College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China 4 Faculty of Science, Huzhou Teachers College, Huzhou, Zhejiang 313000, China Correspondence should be addressed to Gao Hongya, hongya-gao@sohu.com Received 25 September 2009; Accepted 18 March 2010 Academic Editor: Yuming Xing Copyright q 2010 Gao Hongya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Local regularity and local boundedness results for very weak solutions of obstacle problems of the A-harmonic equation divAx, ∇ux  0 are obtained by using the theory of Hodge decomposition, where |Ax, ξ|≈|ξ| p−1 . 1. Introduction and Statement of Results Let Ω be a bounded regular domain in R n , n ≥ 2. By a regular domain we understand any domain of finite measure for which the estimates for the Hodge decomposition in 1.5 and 1.6 are satisfied; see 1. A Lipschitz domain, for example, is a regular domain. We consider the second-order divergence type elliptic equation also called A-harmonic equation or Leray-Lions equation: div A  x, ∇u  x   0, 1.1 where Ax, ξ : Ω × R n → R n is a Carath ´ eodory function satisfying the following conditions: a Ax, ξ,ξ≥α|ξ| p , b |Ax, ξ|≤β|ξ| p−1 , c Ax, 00, 2 Journal of Inequalities and Applications where p>1and0<α≤ β<∞. The prototype of 1.1 is the p-harmonic equation: div  | ∇u | p−2 ∇u   0. 1.2 Suppose that ψ is an arbitrary function in Ω with values in R ∪{±∞},andθ ∈ W 1,r Ω with max{1,p− 1} <r≤ p.Let K r ψ,θ  Ω    v ∈ W 1,r  Ω  : v ≥ ψ a.e., and v − θ ∈ W 1,r 0  Ω   . 1.3 The function ψ is an obstacle and θ determines the boundary values. For any u, v ∈K r ψ,θ Ω, we introduce the Hodge decomposition for |∇v − u| r−p ∇v − u ∈ L r/r−p1 Ω,see1: | ∇  v − u  | r−p ∇  v − u   ∇φ v,u  h v,u , 1.4 where φ v,u ∈ W 1,r/r−p1 0 Ω and h v,u ∈ L r/r−p1 Ω, R n  are a divergence-free vector field, and the following estimates hold:   ∇φ v,u   r/r−p1 ≤ c 1  ∇  v − u   r−p1 r , 1.5  h v,u  r/r−p1 ≤ c 1  p − r   ∇  v − u   r−p1 r , 1.6 where c 1  c 1 n, p is some constant depending only on n and p. Definition 1.1 see 2. A very weak solution to the K r ψ,θ -obstacle problem is a function u ∈ K r ψ,θ Ω such that  Ω  A  x, ∇u  , | ∇  v − u  | r−p ∇  v − u   dx ≥  Ω A  x, ∇u  ,h v,u dx, 1.7 whenever v ∈K r ψ,θ Ω. Remark 1.2. If r  p in Definition 1.1, then h v,u  0 by the uniqueness of the Hodge decomposition 1.4,and1.7 becomes  Ω  A  x, ∇u  , ∇  v − u   dx ≥ 0. 1.8 This is the classical definition for K p ψ,θ -obstacle problem; see 3 for some details of solutions of K p ψ,θ -obstacle problem. Journal of Inequalities and Applications 3 This paper deals with local regularity and local boundedness for very weak solutions of obstacle problems. Local regularity and local boundedness properties are important among the regularity theories of nonlinear elliptic systems; see the recent monograph 4 by Bensoussan and Frehse. Meyers and Elcrat 5 first considered the higher integrability for weak solutions of 1.1 in 1975; see also 6. Iwaniec and Sbordone 1 obtained the regularity result for very weak solutions of the A-harmonic 1.1 by using the celebrated Gehring’s Lemma. The local and global higher integrability of the derivatives in obstacle problem was first considered by Li and Martio 7 in 1994 by using the so-called reverse H ¨ older inequality. Gao et al. 2 gave the definition for very weak solutions of obstacle problem of A-harmonic 1.1 and obtained the local and global higher integrability results. The local regularity results for minima of functionals and solutions of elliptic equations have been obtained in 8. For some new results related to A-harmonic equation, we refer the reader to 9–11. Gao and Tian 12 gave the local regularity result for weak solutions of obstacle problem with the obstacle function ψ ≥ 0. Li and Gao 13 generalized the result of 12 by obtaining the local integrability result for very weak solutions of obstacle problem. The main result of 13 is the following proposition. Proposition 1.3. There exists r 1 with max{1,p− 1} <r 1 <p, such that any very weak solution u to the K r ψ,θ -obstacle problem belongs to L s ∗ loc Ω, s ∗  1/1/s − 1/n, provided that 0 ≤ ψ ∈ W 1,s loc Ω, r<s<n, and r 1 <r<min{p, n}. Notice that in the above proposition we have restricted ourselves to the case r<n, because when r ≥ n, every function in W 1,r loc Ω is trivially in L t loc Ω for every t>1bythe classical Sobolev imbedding theorem. In the first part of this paper, we continue to consider the local regularity theory for very weak solutions of obstacle problem by showing that the condition ψ ≥ 0in Proposition 1.3 is not necessary. Theorem 1.4. There exists r 1 with max{1,p − 1} <r 1 <p, such that any very weak solution u to the K r ψ,θ -obstacle problem belongs to L s ∗ loc Ω, provided that ψ ∈ W 1,s loc Ω, r<s<n, and r 1 <r<min{p, n}. As a corollary of the above theorem, if r  p, that is, if we consider weak solutions of K p ψ,θ -obstacle problem, then we have the following local regularity result. Corollary 1.5. Suppose that ψ ∈ W 1,s loc Ω, 1 <p<s<n. Then a solution u to the K p ψ,θ -obstacle problem belongs to L s ∗ loc Ω. We omit the proof of this corollary. This corollary shows that the condition ψ ≥ 0inthe main result of 12 is not necessary. The second part of this paper considers local boundedness for very weak solutions of K r ψ,θ -obstacle problem. The local boundedness for solutions of obstacle problems plays a central role in many aspects. Based on the local boundedness, we can further study the regularity of the solutions. For the local boundedness results of weak solutions of nonlinear elliptic equations, we refer the reader to 4. In this paper we consider very weak solutions and show that if the obstacle function is ψ ∈ W 1,∞ loc Ω, then a very weak solution u to the K r ψ,θ -obstacle problem is locally bounded. 4 Journal of Inequalities and Applications Theorem 1.6. There exists r 1 with max{1,p − 1} <r 1 <p, such that for any r with r 1 <r< min{p, n} and any ψ ∈ W 1,∞ loc Ω, a very weak solution u to the K r ψ,θ -obstacle problem is locally bounded. Remark 1.7. As far as we are aware, Theorem 1.6 is the first result concerning local boundedness for very weak solutions of obstacle problems. In the remaining part of this section, we give some symbols and preliminary lemmas used in the proof of the main results. If x 0 ∈ Ω and t>0, then B t denotes the ball of radius t centered at x 0 . For a function ux and k>0, let A k  {x ∈ Ω : |ux| >k}, A  k  {x ∈ Ω : ux >k}, A k,t  A k ∩ B t , A  k,t  A  k ∩ B t . Moreover if s<n, s ∗ is always the real number satisfying 1/s ∗  1/s − 1/n.LetT k u be the usual truncation of u at level k>0, that is, T k  u   max { −k, min { k, u }} . 1.9 Let t k umin{u, k}. We recall two lammas which will be used in the proof of Theorem 1.4. Lemma 1.8 see 8. Let u ∈ W 1,r loc Ω, ϕ 0 ∈ L q loc Ω,where1 <r<nand q satisfies 1 <q< n r . 1.10 Assume that the following integral estimate holds:  A k,t | ∇u | r dx ≤ c 0   A k,t ϕ 0 dx   t − τ  −α  A k,t | u | r dx  , 1.11 for every k ∈ N and R 0 ≤ τ<t≤ R 1 ,wherec 0 is a real positive constant that depends only on N, q, r, R 0 ,R 1 , |Ω| and α is a real positive constant. Then u ∈ L qr ∗ loc Ω. Lemma 1.9 see 14. Let fτ be a nonnegative bounded function defined for 0 ≤ R 0 ≤ t ≤ R 1 . Suppose that for R 0 ≤ τ<t≤ R 1 one has f  τ  ≤ A  t − τ  −α  B  θf  t  , 1.12 where A, B, α, θ are nonnegative constants and θ<1. Then there exists a constant c 2  c 2 α, θ, depending only on α and θ, such that for every ρ, R, R 0 ≤ ρ<R≤ R 1 one has f  ρ  ≤ c 2  A  R − ρ  −α  B  . 1.13 We need the following definition. Journal of Inequalities and Applications 5 Definition 1.10 see 15.Afunctionux ∈ W 1,m loc Ω belongs to the class BΩ,γ,m,k 0 , if for all k>k 0 , k 0 > 0andallB ρ  B ρ x 0 , B ρ−ρσ  B ρ−ρσ x 0 , B R  B R x 0 , one has  A  k,ρ−ρσ | ∇u | m dx ≤ γ  σ −m ρ −m  A  k,ρ  u − k  m dx     A  k,ρ     , 1.14 for R/2 ≤ ρ − ρσ<ρ<R, m<n, where |A  k,ρ | is the n-dimensional Lebesgue measure of the set A  k,ρ . We recall a lemma from 15 which will be used in the proof of Theorem 1.6. Lemma 1.11 see 15. Suppose that ux is an arbitrary function belonging to the class BΩ,γ,m,k 0  and B R ⊂⊂ Ω. Then one has max B R/2 u  x  ≤ c, 1.15 in which the constant c is determined only by the quantities γ,m,k 0 ,R,∇u m 1 . 2. Local Regularity Proof of Theorem 1.4. Let u be a very weak solution to the K r ψ,θ -obstacle problem. By Lemma 1.8,itissufficient to prove that u satisfies the inequality 1.11 with α  r.Let B R 1 ⊂⊂ Ω and 0 <R 0 ≤ τ<t≤ R 1 be arbitrarily fixed. Fix a cut-off function φ ∈ C ∞ 0 B R 1  such that supp φ ⊂ B t , 0 ≤ φ ≤ 1,φ 1inB τ ,   ∇φ   ≤ 2  t − τ  −1 . 2.1 Consider the function v  u − T k  u  − φ r  u − ψ k  , 2.2 where T k u is the usual truncation of u at level k ≥ 0 defined in 1.9 and ψ k  max{ψ, T k u}. Now v ∈K r ψ−T k u,θ−T k u Ω; indeed, since u ∈K r ψ,θ Ω and φ ∈ C ∞ 0 Ω, then v −  θ − T k  u   u − θ − φ r  u − ψ k  ∈ W 1,r 0  Ω  , v −  ψ − T k  u    u − ψ − φ r  u − ψ k  ≥  1 − φ r  u − ψ  ≥ 0, 2.3 6 Journal of Inequalities and Applications a.e. in Ω.Let E  v, u     φ r ∇u   r−p φ r ∇u  | ∇  v − u  T k  u  | r−p ∇  v − u  T k  u  , 2.4 By an elementary inequality 16, Page 271, 4.1,   | X | −ε X − | Y | −ε Y   ≤ 2 ε 1  ε 1 − ε | X − Y | 1−ε ,X,Y∈ R n , 0 ≤ ε<1, ∇v  ∇  u − T k  u  − φ r ∇  u − ψ k  − rφ r−1 ∇φ  u − ψ k  , 2.5 one can derive that | E  v, u  | ≤ 2 p−r p − r  1 r − p  1    φ r ∇ψ k − rφ r−1 ∇φ  u − ψ k     r−p1 . 2.6 We get from the definition of Ev, u that  A k,t  A  x, ∇u  ,   φ r ∇u   r−p φ r ∇u  dx   A k,t  A  x, ∇u  ,E  v, u   dx −  A k,t  A  x, ∇u  , | ∇  v − u  T k  u  | r−p ∇  v − u  T k  u   dx   A k,t  A  x, ∇u  ,E  v, u   dx −  A k,t  A  x, ∇u  , | ∇  v − u  | r−p ∇  v − u   dx. 2.7 Now we estimate the left-hand side of 2.7. By condition a we have  A k,t A  x, ∇u  ,   φ r ∇u   r−p φ r ∇ud ≥  A k,τ A  x, ∇u  , | ∇u | r−p ∇udx ≥ α  A k,τ | ∇u | r dx. 2.8 Since u − T k u,v ∈K r ψ−T k u,θ−T k u Ω, then using the Hodge decomposition 1.4,weget | ∇  v − u  T k  u  | r−p ∇  v − u  T k  u   ∇φ  h, 2.9 and by 1.6 we have  h  r/r−p1 ≤ c 1  p − r   ∇v − u  T k u  r−p1 r . 2.10 Journal of Inequalities and Applications 7 Thus we derive, by Definition 1.1,that  Ω  A  x, ∇  u − T k  u  , | ∇  v − u  T k  u  | r−p ∇  v − u  T k  u   dx ≥  Ω  A  x, ∇  u − T k  u  ,h  dx. 2.11 This means, by condition c,that  A k,t A  x, ∇u  , | ∇  v − u  | r−p ∇  v − u  dx ≥  A k,t  A  x, ∇u  ,h  dx. 2.12 Combining the inequalities 2.7, 2.8,and2.12,andusingH ¨ older’s inequality and condition b,weobtain α  A k,τ | ∇u | r dx ≤  A k,t  A  x, ∇u  ,E  v, u   dx −  A k,t  A  x, ∇u  ,h  dx ≤ β 2 p−r  p − r  1  r − p  1  A k,t | ∇u | p−1    φ r ∇ψ k − rφ r−1 ∇φ  u − ψ k     r−p1 dx  β  A k,t | ∇u | p−1 | h | dx ≤ β 2 p−r  p − r  1  r − p  1  A k,t | ∇u | p−1   φ r ∇ψ k   r−p1 dx  β 2 p−r  p − r  1  r − p  1  A k,t | ∇u | p−1    rφ r−1 ∇φ  u − ψ k     r−p1 dx  β  A k,t | ∇u | p−1 | h | dx ≤ β 2 p−r  p − r  1  r − p  1   A k,t | ∇u | r dx  p−1/r   A k,t   ∇ψ k   r dx  r−p1/r  β 2 p−r  p − r  1  r − p  1   A k,t | ∇u | r dx  p−1/r ×   A k,t    rφ r−1 ∇φ  u − ψ k     r dx  r−p1/r  β   A k,t | ∇u | r dx  p−1/r   A k,t | h | r/r−p1 dx  r−p1/r . 2.13 8 Journal of Inequalities and Applications Denote c 3  c 3 p, r2 p−r p − r  1/r − p  1. It is obvious that if r is sufficiently close to p, then c 3 p, r ≤ 2. By 2.10 and Young’s inequality ab ≤ εa p   c 4  ε, p  b p , 1 p  1 p   1,a,b≥ 0,ε≥ 0,p≥ 1, 2.14 we can derive that α  A k,τ | ∇u | r dx ≤ βc 3  p, r  ε  A k,t | ∇u | r dx  βc 3  p, r  c 4  ε, p   A k,t   ∇ψ k   r dx  βc 3  p, r  ε  A k,t | ∇u | r dx  βc 3  p, r  c 4  ε, p   A k,t    rφ r−1 ∇φ  u − ψ k     r dx  βc 1 ε  p − r   A k,t | ∇u | r dx  βc 1 c 4  ε, p  p − r   Ω | ∇  v − u  T k  u  | r dx ≤ βε  2c 3  p, r   c 1  p − r   A k,t | ∇u | r dx  βc 3  p, r  c 4  ε, p   A k,t   ∇ψ k   r dx  βc 3  p, r  c 4  ε, p   A k,t    rφ r−1 ∇φ  u − ψ k     r dx  βc 1 c 4  ε, p  p − r   Ω | ∇  v − u  T k  u  | r dx. 2.15 By the equality ∇v  ∇  u − T k  u  − φ r ∇  u − ψ k  − rφ r−1 ∇φ  u − ψ k  , 2.16 and v − u  T k u0forx ∈ Ω \ A k,t , then we have  Ω | ∇  v − u  T k  u  | r dx   A k,t    φ r ∇  u − ψ k   rφ r−1 ∇φ  u − ψ k     r dx ≤ 2 r−1   A k,t | ∇u | r dx   A k,t   ∇ψ k   r dx   A k,t    rφ r−1 ∇φ  u − ψ k     r dx  . 2.17 Journal of Inequalities and Applications 9 Finally we obtain that  A k,τ | ∇u | r dx ≤ βε  2c 3  p, r   c 1  p − r   2 r−1 βc 1 c 4  ε, p  p − r  α  A k,t | ∇u | r dx  βc 3  p, r  c 4  ε, p   2 r−1 βc 1 c 4  ε, p  p − r  α  A k,t   ∇ψ k   r dx  βc 3  p, r  c 4  ε, p   2 r−1 βc 1 c 4  ε, p  p − r  α  A k,t    rφ r−1 ∇φ  u − ψ k     r dx ≤ βε  2c 3  p, r   c 1  p − r   2 p−1 βc 1 c 4  ε, p  p − r  α  A k,t | ∇u | r dx  βc 3  p, r  c 4  ε, p   2 p−1 βc 1 c 4  ε, p  p − r  α  A k,t   ∇ψ   r dx  βc 3  p, r  c 4  ε, p   2 p−1 βc 1 c 4  ε, p  p − r  α 2 p p  t − τ  r  A k,t | u | r dx. 2.18 The last i nequality holds since |u − ψ k |≤|u| a.e. in A k,t . Now we want to eliminate the first term in the right-hand side containing ∇u. Choose ε small enough and r sufficiently close to p such that θ  βε  2c 3  p, r   c 1  p − r   2 p−1 βc 1 c 4  ε, p  p − r  α < 1, 2.19 and let ρ,R be arbitrarily fixed with R 0 ≤ ρ<R≤ R 1 .Thus,from2.18, we deduce that for every τ and t such that ρ ≤ τ<t≤ R, we have  A k,τ | ∇u | r dx ≤ θ  A k,t | ∇u | r dx  c 5 α  A k,R   ∇ψ   r dx  c 6 α  t − τ  r  A k,R | u | r dx, 2.20 where c 5  βc 3 p, rc 4 ε, p2 p−1 βc 1 c 4 ε, pp − r with ε and r fixed to satisfy 2.19,and c 6  2 p pc 5 . Applying Lemma 1.9 in 2.20 we conclude that  A k,ρ | ∇u | r dx ≤ c 2 c 5 α  A k,R   ∇ψ   r dx  c 2 c 6 α  R − ρ  r  A k,R | u | r dx, 2.21 where c 2 is the constant given by Lemma 1.9.Thusu satisfies inequality 1.11 with ϕ 0  |∇ψ| r and α  r. Theorem 1.4 follows from Lemma 1.8. 10 Journal of Inequalities and Applications 3. Local Boundedness Proof of Theorem 1.6. Let u be a very weak solution to the K r ψ,θ -obstacle problem. Let B R 1 ⊂⊂ Ω and R 1 /2 ≤ τ<t≤ R 1 be arbitrarily fixed. Fix a cut-off function φ ∈ C ∞ 0 B R 1  such that supp φ ⊂ B t , 0 ≤ φ ≤ 1,φ 1inB τ ,   ∇φ   ≤ 2  t − τ  −1 . 3.1 Consider the function v  u − t k  u  − φ r  u − max  ψ, t k  u   , 3.2 where t k umin{u, k}.Nowv ∈K r ψ−t k u,θ−t k u ; indeed, since u ∈K r ψ,θ Ω and φ ∈ C ∞ 0 Ω, then v −  θ − t k  u   u − θ − φ r  u − max  ψ, t k  u   ∈ W 1,r 0  Ω  , v −  ψ − t k  u    u − ψ − φ r  u − max  ψ, t k  u   ≥  1 − φ r  u − ψ  ≥ 0 3.3 a.e. in Ω. As in the proof of Theorem 1.4,weobtain  A  k,τ | ∇u | r dx ≤ βε  2c 3  p, r   c 1  p − r   2 r−1 βc 1 c 4  ε, p  p − r  α  A  k,t | ∇u | r dx  βc 3 c 4  ε, p   2 r−1 βc 1 c 4  ε, p  p − r  α  A  k,t   ∇ max  ψ, t k  u     r dx  βc 3  p, r  c 4  ε, p   2 r−1 βc 1 c 4  ε, p  p − r  α ×  A  k,t    rφ r−1 ∇φu − max{ψ, t k u}    r dx ≤ βε  2c 3  p, r   c 1  p − r   2 r−1 βc 1 c 4  ε, p  p − r  α  A  k,t | ∇u | r dx  βc 3 c 4  ε, p   2 r−1 βc 1 c 4  ε, p  p − r  α  A  k,t   ∇ψ   r dx  βc 3  p, r  c 4  ε, p   2 r−1 βc 1 c 4  ε, p  p − r  α 2 p p  t − τ  r  A  k,t | u − k | r dx. 3.4 Choose ε small enough and r 1 sufficiently close to p such that 2.19 holds. Let ρ, R be arbitrarily fixed with R 1 /2 ≤ ρ<R≤ R 1 .Thusfrom3.4 we deduce that for every τ and t such that R 1 /2 ≤ τ<t≤ R 1 , we have  A  k,τ | ∇u | r dx ≤ θ  A  k,t | ∇u | r dx  c 5 α  A  k,R   ∇ψ   r dx  c 6 α  t − τ  r  A  k,R | u − k | r dx. 3.5 [...]... anisotropic equations,” Journal of Inequalities and Applications, vol 2008, Article ID 835736, 11 pages, 2008 12 Journal of Inequalities and Applications 12 H Gao and H Tian, Local regularity result for solutions of obstacle problems,” Acta Mathematica Scientia B, vol 24, no 1, pp 71–74, 2004 13 J Li and H Gao, Local regularity result for very weak solutions of obstacle problems,” Radovi Matematiˇ ki,... angewandte Mathematik, vol 454, pp 143–161, 1994 2 H Y Gao, M Wang, and H L Zhao, Very weak solutions for obstacle problems of the A-harmonic equation,” Journal of Mathematical Research and Exposition, vol 24, no 1, pp 159–167, 2004 Chinese 3 J Heinonen, T Kilpel¨ inen, and O Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, a Oxford Mathematical Monographs, The Clarendon Press, Oxford,... 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The local regularity results for minima of functionals and solutions. solutions of K p ψ,θ -obstacle problem. Journal of Inequalities and Applications 3 This paper deals with local regularity and local boundedness for very weak solutions of obstacle problems. Local regularity. Corporation Journal of Inequalities and Applications Volume 2010, Article ID 878769, 12 pages doi:10.1155/2010/878769 Research Article Local Regularity and Local Boundedness Results for Very Weak Solutions of Obstacle