RESEARCH Open Access A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight Gao Hongya 1* and Qiao Jinjing 1,2 * Correspondence: hongya- gao@sohu.com 1 College of Mathematics and Computer Science, Hebei University, Baoding 071002, People’s Republic of China Full list of author information is available at the end of the article Abstract This paper gives a Caccioppoli-type estimate for very weak solutions to obstacle problems of the A -harmonic equation divA ( x, ∇u ) =0 with |A ( x, ξ ) |≈w ( x ) |ξ| p− 1 , where 1 <p < ∞ and w(x) be a Muckenhoupt A 1 weight. Mathematics Subject Classificat ion (2000) 35J50, 35J60 Keywords: Caccioppoli-type estimate, very weak solution, obstacle problem, Mucken- houpt weight, A-harmonic equation 1 Introduction Let w be a locally integrable non-negative function in R n and assume th at 0 <w < ∞ almost everywhere. We say that w belongs to the Muckenhoupt class A p ,1<p < ∞,or that w is an A p weight, if there is a constant A p (w) such that sup B 1 |B| B wdx 1 |B| B w 1/(1−p) dx p−1 = A p (w) < ∞ (1:1) for all balls B in R n .Wesaythatw belongs to A 1 ,orthatw is an A 1 weigh t, if there is a constant A 1 (w) such that 1 |B| B wdx ≤ A 1 (w)essinf B w for all balls B in R n . As customary, μ stands for the measure whose Radon-Nikodym derivative w is μ(E)= E wdx . It is well know n that A 1 ⊂ A p whenever p >1,see[1].Wesaythataweightw is doubling if there is a constant C > 0 such that μ ( 2B ) ≤ Cμ ( B ) whenever B ⊂ 2B are concentric balls in R n ,where2B is the ball with the same cen- ter as B an d with radius twice that of B. Given a measurable subset E of R n , we will denote by L p (E, w), 1 <p < ∞, the Banach spa ce of all measurable functions f defined on E for which Hongya and Jinjing Journal of Inequalities and Applications 2011, 2011:58 http://www.journalofinequalitiesandapplications.com/content/2011/1/58 © 2011 Hongya and Jinjing; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unres tricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | |f || L p (E,w) = ⎛ ⎝ E |f (x)| p w(x)dx ⎞ ⎠ 1 p < ∞ . The weighted Sobolev class W 1,p (E, w) consists of all functions f, and its first general- ized derivatives belong to L p ( E, w). The s ymbols L p l oc (E, w ) and W 1 ,p l oc (E, w ) are self- explanatory. If x 0 Î Ω and t > 0, then B t denotes the ball of radius t centered at x 0 . For the func- tion u(x)andk >0,letA k ={x Î Ω :|u(x)| >k}, A k,t = A k ∩ B t .LetT k (u) be the usual truncation of u at level k > 0, that is T k ( u ) =max{−k, min{k, u}} . Let Ω be a bounded regular domain in R n , n ≥ 2. By a regular domain, we under- stand any domain of finite measure for which the estimates for th e Hodge decomposi- tion in (2.1) and (2.2) are satisfied. A Lips chit z domain, for example, is regular. We consider the second-order degenerate elliptic equation (also called A -harmonic equa- tion or Leray-Lions equation) divA ( x, ∇u ) = 0 (1:2) where A ( x, ξ ) : × R n → R n is a carathéodory function satisfying the following assumptions 1. A ( x, ξ ) , ξ ≥αw ( x ) |ξ| p , 2. |A ( x, ξ ) |≤βw ( x ) |ξ| p− 1 , where 0 <a ≤ b < ∞, w Î A 1 and w ≥ k 0 > 0. Suppose ψ is any function in Ω with values in the extended reals [-∞,+∞] and that θ Î W 1,r (Ω, w), max{1, p -1} <r ≤ p. Let K r ψ ,θ = K r ψ ,θ (, w)={v ∈ W 1,r (, w):v ≥ ψ ,a.e.x ∈ and v − θ ∈ W 1,r 0 (, w)} . The function ψ is an obstacle, and θ determines the boundary values. We introduce the Hodge decomposition for |∇ ( v − u ) | r−p ∇ ( v − u ) ∈ L r r−p+1 ( , w ) , from Lemma 1 in Section 2, |∇ ( v − u ) | r−p ∇ ( v − u ) = ∇ϕ + H (1:3) and the following estimate holds H L r r − p +1 (,w) ≤ cA p (w) γ |r − p|∇(v − u) r−p+ 1 L r (,w) . (1:4) Definition 1 A very weak solution to the K r ψ , θ -obstacle problem is a function u ∈ K r ψ ,θ (, w ) such that A(x, ∇u), |∇(v − u)| r−p ∇(v − u)dx ≥ A(x, ∇u), Hd x (1:5) whenever v ∈ K r ψ ,θ (, w ) and H comes from the Hodge decomposition (1.3). Hongya and Jinjing Journal of Inequalities and Applications 2011, 2011:58 http://www.journalofinequalitiesandapplications.com/content/2011/1/58 Page 2 of 7 The local and global higher integrability of the derivatives in obstacle problems with w(x) ≡ 1 was first considered by Li and Mar tio [2] in 1994, using the s o-called reverse Hölder inequality. Gao and Tian [3] gave a local regularity result fo r weak solutions to obstacle problem in 2004. Recently, regularity theory for very weak solutions of the A -harmonic equations with w(x) ≡ 1 have been considere d [4], and the regularity the- ory for very solutions of obstacle problems with w(x) ≡ 1 have been explored in [5]. This paper gives a Caccioppoli-type estimate for solutions to obstacle problems with weight, which is closely related to the local regularity theory for very wea k solutions of the A -harmonic equation (1.2). Theorem There exists r 1 Î (p -1,p) such that for arbitrary ψ ∈ W 1,p l oc (, w ) and r 1 <r <p, a solution u to the K r ψ ,θ -obstacle problem with weight w(x) Î A 1 satisfies the follow- ing Caccioppoli-type estimate A k,ρ |∇u| r dμ ≤ C ⎡ ⎢ ⎣ A k,R |∇ψ| r dμ + 1 (R − ρ) r A k,R |u| r dμ ⎤ ⎥ ⎦ where 0<r <R <+∞ and C is a constant depends only on n, p and b/a. 2 Preliminary Lemmas The following lemma comes from [6] which is a Hod ge decomposition in weighted spaces. Lemma 1 Let Ω be a regular domain of R n and w(x) be an A 1 weight. If u ∈ W 1,p−ε 0 (, w ) ,1<p < ∞,-1<ε <p -1,then there exist ϕ ∈ W 1, p−ε 1−ε 0 (, w ) and a divergence-free vector field H ∈ L p−ε 1−ε ( , w ) such that | ∇u| − ε ∇u = ∇ϕ + H and ∇ϕ L p − ε 1 − ε ( ,w ) ≤ cA p (w) γ ∇u 1 −ε L p−ε (,w) (2:1) H L p − ε 1 − ε ( ,w ) ≤ cA p (w) γ |ε|∇u 1−ε L p−ε (,w ) (2:2) where g depends only on p. We also need the following lemma in the proof of the main theorem. Lemma 2 [7]Let f(t) be a non-negative bounded function defined for 0 ≤ T 0 ≤ t ≤ T 1 . Suppose that for T 0 ≤ t <s ≤ T 1 , we have f ( t ) ≤ A ( s − t ) −α + B + θ f ( s ), where A, B, a, θ are non-negative constants and θ <1.Then, there exist a constant c, depending only on a and θ, such that for every r, R, T 0 ≤ r <R ≤ T 1 we have f ( ρ ) ≤ c[A ( R − ρ ) − α + B] . Hongya and Jinjing Journal of Inequalities and Applications 2011, 2011:58 http://www.journalofinequalitiesandapplications.com/content/2011/1/58 Page 3 of 7 3 Proof of the main theorem Let u be a very weak solution to the K r ψ ,θ -obstacle problem. Let B R 1 ⊂⊂ and 0 <R 0 ≤ τ <t ≤ R 1 be arbitrarily fixed. Fix a cut-off function φ ∈ C ∞ 0 (B t ) such that suppφ ⊂ B t ,0≤ φ ≤ 1, φ =1inB τ and |∇φ|≤2 ( t − τ ) −1 . Consider the function v = u − T k (u) − φ r (u − ψ + k ) , where T k (u) is the usual truncation of u at the level k defined in Section 1 and ψ + k =max{ψ, T k (u) } . Now v ∈ K r ψ−T k ( u ) ,θ−T k ( u ) (, w ) . Indeed, v − (θ − T k (u)) = u − θ − φ r (u − ψ + k ) ∈ W 1,r 0 (, w ) since φ ∈ C ∞ 0 ( ) and v − (ψ − T k (u)) = (u − ψ) − φ r (u − ψ + k ) ≥ (1 − φ r )(u − ψ) ≥ 0 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 )). (3:1) From an elementary formula [[8], (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 and ∇ v = ∇(u − T k (u)) − φ r ∇(u − ψ + k ) − rφ r−1 ∇φ(u − ψ + k ) , we can derive that |E(v, u)|≤2 p−r p − r +1 r − p +1 |φ r ∇u − φ r ∇(u − ψ + k ) − rφ r−1 ∇φ(u − ψ + k )| r−p+1 . (3:2) From (3.1), we get that A k,t A(x, ∇u), |φ r ∇u| r−p φ r ∇udx = A k,t A(x, ∇u), E(v, u)dx − A k , t A(x, ∇u), |∇(v − u)| r−p ∇(v − u)dx . (3:3) Now we estimate the left-hand side of (3.3), A k , t A(x, ∇u), |φ r ∇u| r−p φ r ∇udx ≥ A k , τ A(x, ∇u), |∇u| r−p ∇udx ≥ α A k , τ |∇u| r dμ . (3:4) Using (1.3), we get | ∇ ( v − u + T k ( u )) | r−p ∇ ( v − u + T k ( u )) = ∇ϕ + H (3:5) and (1.4) yields H L r r − p +1 ( ,w ) ≤ cA p (w) γ |r − p|∇(v − u + T k (u)) r−p+ 1 L r (,w) . (3:6) Since u - T k (u) is a very weak solution to the K r ψ−T k ( u ) ,θ−T k ( u ) -obstacle problem, we derive, by Hongya and Jinjing Journal of Inequalities and Applications 2011, 2011:58 http://www.journalofinequalitiesandapplications.com/content/2011/1/58 Page 4 of 7 Definition 1, that A(x, ∇(u−T k (u))), |∇(v−u+T k (u))| r−p ∇(v−u+T k (u))dx ≥ A(x, ∇(u−T k (u))), Hd x that is A k , t A(x, ∇u), |∇(v − u)| r−p ∇(v − u)dx ≥ A k , t A(x, ∇u), Hdx . (3:7) Combining the inequalities (3.3), (3.4) and (3.7), we obtain α A k,τ |∇u| r dμ ≤ A k,t A(x, ∇ u), E(v, u)d x − A k,t A(x, ∇ u), Hdx ≤ 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 dμ + β A k,t |∇u| p−1 |H|dμ ≤ β 2 p−r (p − r +1) r − p +1 A k,t |∇u| p−1 |φ r ∇ψ | r−p+1 dμ + 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 dμ + β A k,t |∇u| p−1 |H|dμ ≤ β 2 p−r (p − r +1) r − p +1 ⎛ ⎜ ⎝ A k,t |∇u| r dμ ⎞ ⎟ ⎠ p−1 r ⎛ ⎜ ⎝ A k,t |∇ψ | r dμ ⎞ ⎟ ⎠ r−p+1 r + β 2 p−r (p − r +1) r − p +1 ⎛ ⎜ ⎝ A k,t |∇u| r dμ ⎞ ⎟ ⎠ p−1 r ⎛ ⎜ ⎝ A k,t |rφ p−1 ∇φ(u − ψ + k )| r dμ ⎞ ⎟ ⎠ r−p+1 r + β ⎛ ⎜ ⎝ A k,t |∇u| r dμ ⎞ ⎟ ⎠ p−1 r ⎛ ⎜ ⎝ A k,t |H| r r − p +1 dμ ⎞ ⎟ ⎠ r−p+1 r . Let c 1 = 2 p −r (p−r+1) r− p +1 , by (3.6) and Young’s inequality ab ≤ εa p + c 2 (ε, p)b p , 1 p + 1 p =1, a, b ≥ 0, ε ≥ 0, p ≥ 1 , we can derive that α A k,τ |∇u| r dμ ≤βc 1 ε A k,t |∇u| r dμ + βc 1 c 2 (ε, p) A k,t |∇ψ| r dμ + βc 1 ε A k,t |∇u| r dμ + βc 1 c 2 (ε, p) A k,t |rφ r−1 ∇φ(u − ψ + k )| r d μ + βcA p (w) γ (p − r)ε A k,t |∇u| r dμ + βcA p (w) γ (p − r)c 2 (ε, p) |∇(v − u + T k (u))| r dμ, where c is the constant given by Lemma 1. Sin ce v - u + T k (u)=0onΩ\A k,t ,bythe equality Hongya and Jinjing Journal of Inequalities and Applications 2011, 2011:58 http://www.journalofinequalitiesandapplications.com/content/2011/1/58 Page 5 of 7 ∇ v = ∇(u − T k (u)) − φ r ∇(u − ψ + k ) − rφ r−1 ∇φ(u − ψ + k ) , we obtain that |∇(v − u + T k (u))| r dμ = A k,t |∇(v − u)| r dμ = A k,t |φ r ∇(u − ψ + k )+rφ r−1 ∇φ(u − ψ + k )| r dμ ≤ 2 r−1 A k,t |∇(u − ψ + k )| r dμ +2 r−1 r Ak,t |∇φ(u − ψ + k )| r dμ ≤ 2 2r−2 A k , t |∇u| r dμ +2 2r−2 A k , t |∇ψ| r dμ + r2 2r−2 A k , t |u r | (t − τ) r dμ . Finally, we obtain A k,τ |∇u| r dμ ≤ β(2c 1 + cA p (w) γ (p − r))ε + βcA p (w) γ c 2 (ε, p)2 2 r− 2 (p − r) α A k,t |∇u| r d μ + βc 1 c 2 (ε, p)+2 2r−2 βcA p (w) γ c 2 (ε, p)(p − r) α A k,t |∇ψ| r dμ + r βc 1 c 2 (ε, p)+2 2r−1 βcA p (w) γ c 2 (ε, p)(p − r) α A k , t |u| r (t − τ ) r dμ. (3:8) Now we want to eliminate the first term in the right-hand side c ontaining ∇u. Choosing ε and r 1 such that η = β(2c 1 + cA p (w) γ (p − r))ε + βcA p (w) γ c 2 (ε, p)2 2r−2 (p − r) α < 1 and let r, R be arbitrarily fixed with R 0 ≤ r <R ≤ R 1 . Thus, from (3.8), we deduce that for every t and τ such that r ≤ τ <t ≤ R, we have A k , τ |∇u| r dμ ≤ η A k , t |∇u| r dμ + c 3 α A k , t |∇ψ|dμ + c 4 α(t − τ) r A k , t |u| r dμ , (3:9) where c 3 = βc 1 c 2 (ε, p)+2 2r−2 βcA p (w) γ c 2 (ε, p)(p − r ) and c 4 = rβc 1 c 2 (ε, p)+r2 2r−1 βcA p (w) γ c 2 (ε, p)(p − r) . Applying Lemma 2 in (3.9), we conclude that A k, ρ |∇u| r dμ ≤ cc 3 α A k,R |∇ψ| r dμ + cc 4 α(R − ρ) r A k,R |u| r dμ, where c is the constant given by Lemma 2. This ends the proof of the main theorem. Hongya and Jinjing Journal of Inequalities and Applications 2011, 2011:58 http://www.journalofinequalitiesandapplications.com/content/2011/1/58 Page 6 of 7 Acknowledgements The authors would like to thank the referee of this paper for helpful suggestions. Research supported by NSFC (10971224) and NSF of Hebei Province (A2011201011). Author details 1 College of Mathematics and Computer Science, Hebei University, Baoding 071002, People’s Republic of China 2 College of Mathematics and Computer Science, Hunan Normal University, Changsha 410082, People’s Republic of China Authors’ contributions GH gave Definition 1. QJ found Lemmas 1 and 2. Theorem 1 was proved by both authors. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 3 March 2011 Accepted: 17 September 2011 Published: 17 September 2011 References 1. Heinonen, J, Kilpeläinen, T, Martio, O: Nonlinear potential theory of degenerate elliptic equations. Dover Publications, New York (2006) 2. Li, GB, Martio, O: Local and global integrability of gradients in obstacle problems. Ann Acad Sci Fenn Ser A I Math. 19, 25–34 (1994) 3. 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Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Hongya and Jinjing Journal of Inequalities and Applications 2011, 2011:58 http://www.journalofinequalitiesandapplications.com/content/2011/1/58 Page 7 of 7 . RESEARCH Open Access A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight Gao Hongya 1* and Qiao Jinjing 1,2 * Correspondence: hongya- gao@sohu.com 1 College of Mathematics. inequality. Gao and Tian [3] gave a local regularity result fo r weak solutions to obstacle problem in 2004. Recently, regularity theory for very weak solutions of the A -harmonic equations with. Sbordone, C: Weak minima of variational integrals. J Reine Angew Math. 454, 143–161 (1994) 5. Li, J, Gao, HY: Local regularity result for very weak solutions of obstacle problems. Radovi Math. 12,19–26