Open Access

A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight

Journal of Inequalities and Applications20112011:58

https://doi.org/10.1186/1029-242X-2011-58

Received: 3 March 2011

Accepted: 17 September 2011

Published: 17 September 2011

Abstract

This paper gives a Caccioppoli-type estimate for very weak solutions to obstacle problems of the A -harmonic equation div A ( x , u ) = 0 with | A ( x , ξ ) | w ( x ) | ξ | p - 1 , where 1 < p < ∞ and w(x) be a Muckenhoupt A1 weight.

Mathematics Subject Classification (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 that 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 w d x 1 | B | B w 1 ( 1 - p ) d x p - 1 = A p ( w ) <
(1.1)
for all balls B in R n . We say that w belongs to A1, or that w is an A1 weight, if there is a constant A1(w) such that
1 | B | B w d x A 1 ( w ) essin f 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 w d x .
It is well known that A1 A p whenever p > 1, see [1]. We say that a weight w is doubling if there is a constant C > 0 such that
μ ( 2 B ) C μ ( B )
whenever B 2B are concentric balls in R n , where 2B is the ball with the same center as B and 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 space of all measurable functions f defined on E for which
| | f | | L p ( E , w ) = ( E | f ( x ) | p w ( x ) d x ) 1 p < .

The weighted Sobolev class W1,p(E, w) consists of all functions f, and its first generalized derivatives belong to L p (E, w). The symbols L loc p ( E , w ) and W loc 1 , p ( E , w ) are self-explanatory.

If x0 Ω and t > 0, then B t denotes the ball of radius t centered at x0. For the function u(x) and k > 0, let A k = {x Ω : |u(x)| > k}, Ak,t= A k B t . Let T 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 understand any domain of finite measure for which the estimates for the Hodge decomposition in (2.1) and (2.2) are satisfied. A Lipschitz domain, for example, is regular. We consider the second-order degenerate elliptic equation (also called A -harmonic equation or Leray-Lions equation)
div A ( x , u ) = 0
(1.2)
where A ( x , ξ ) : Ω × R n R n is a carathéodory function satisfying the following assumptions
  1. 1.

    A ( x , ξ ) , ξ α w ( x ) | ξ | p ,

     
  2. 2.

    | A ( x , ξ ) | β w ( x ) | ξ | p - 1 ,

     
where 0 < αβ < ∞, w A1 and wk0 > 0. Suppose ψ is any function in Ω with values in the extended reals [-∞, +∞] and that θ W1,r(Ω, w), max{1, p -1} < rp. Let
K ψ , θ r = K ψ , θ r ( Ω , w ) = { v W 1 , r ( Ω , w ) : v ψ , a.e. x Ω and v - θ W 0 1 , r ( Ω , 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 ( Ω , 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 ( Ω , w ) c A p ( w ) γ | r - p | ( v - u ) L r ( Ω , w ) r - p + 1 .
(1.4)
Definition 1 A very weak solution to the K ψ , θ r -obstacle problem is a function u K ψ , θ r ( Ω , w ) such that
Ω A ( x , u ) , | ( v - u ) | r - p ( v - u ) d x Ω A ( x , u ) , H d x
(1.5)

whenever v K ψ , θ r ( Ω , w ) and H comes from the Hodge decomposition (1.3).

The local and global higher integrability of the derivatives in obstacle problems with w(x) ≡ 1 was first considered by Li and Martio [2] in 1994, using the so-called reverse Hölder inequality. Gao and Tian [3] gave a local regularity result for 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 considered [4], and the regularity theory 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 weak solutions of the A -harmonic equation (1.2).

Theorem There exists r1 (p - 1, p) such that for arbitrary ψ W loc 1 , p ( Ω , w ) and r1 < r < p, a solution u to the K ψ , θ r -obstacle problem with weight w(x) A1 satisfies the following Caccioppoli-type estimate
A k , ρ | u | r d μ C A k , R | ψ | r d μ + 1 ( R - ρ ) r A k , R | u | r d μ

where 0 < ρ < R < +∞ and C is a constant depends only on n, p and β/α.

2 Preliminary Lemmas

The following lemma comes from [6] which is a Hodge decomposition in weighted spaces.

Lemma 1 Let Ω be a regular domain of R n and w(x) be an A1 weight. If u W 0 1 , p - ε ( Ω , w ) , 1 < p < ∞, -1 < ε < p - 1, then there exist φ W 0 1 , p - ε 1 - ε ( Ω , w ) and a divergence-free vector field H L p - ε 1 - ε ( Ω , w ) such that
| u | - ε u = φ + H
and
φ L p - ε 1 - ε ( Ω , w ) c A p ( w ) γ u L p - ε ( Ω , w ) 1 - ε
(2.1)
H L p - ε 1 - ε ( Ω , w ) c A p ( w ) γ | ε | u L p - ε ( Ω , w ) 1 - ε
(2.2)

where γ 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 ≤ T0tT1. Suppose that for T0t < sT1, we have
f ( t ) A ( s - t ) - α + B + θ f ( s ) ,
where A, B, α, θ are non-negative constants and θ < 1. Then, there exist a constant c, depending only on α and θ, such that for every ρ, R, T0ρ < RT1 we have
f ( ρ ) c [ A ( R - ρ ) - α + B ] .

3 Proof of the main theorem

Let u be a very weak solution to the K ψ , θ r -obstacle problem. Let B R 1 Ω and 0 < R0τ < tR1 be arbitrarily fixed. Fix a cut-off function ϕ C 0 ( B t ) such that
supp ϕ B t , 0 ϕ 1 , ϕ = 1 in B 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 ψ - T k ( u ) , θ - T k ( u ) r ( Ω , w ) . Indeed,
v - ( θ - T k ( u ) ) = u - θ - ϕ r ( u - ψ k + ) W 0 1 , r ( Ω , w )
since ϕ C 0 ( Ω ) 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 + ) - r ϕ 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 + ) - r ϕ 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 u d x = A k , t A ( x , u ) , E ( v , u ) d x (1) - A k , t A ( x , u ) , | ( v - u ) | r - p ( v - u ) d x . (2) (3) 
(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 u d x A k , τ A ( x , u ) , | u | r - p u d x α A k , τ | u | r d μ .
(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 ( Ω , w ) c A p ( w ) γ | r - p | ( v - u + T k ( u ) ) L r ( Ω , w ) r - p + 1 .
(3.6)

Since u - T k (u) is a very weak solution to the K ψ - T k ( u ) , θ - T k ( u ) r -obstacle problem, we derive, by

Definition 1, that
Ω A ( x , ( u - T k ( u ) ) ) , | ( v - u + T k ( u ) ) | r - p ( v - u + T k ( u ) ) d x Ω A ( x , ( u - T k ( u ) ) ) , H d x
that is
A k , t A ( x , u ) , | ( v - u ) | r - p ( v - u ) d x A k , t A ( x , u ) , H d x .
(3.7)
Combining the inequalities (3.3), (3.4) and (3.7), we obtain
α A k , τ | u | r d μ A k , t A ( x , u ) , E ( v , u ) d x A k , t A ( x , u ) , H d x β 2 p r ( p r + 1 ) r p + 1 A k , t | u | p 1 | ϕ r ψ k + r ϕ r 1 ϕ ( u ψ k + ) | r p + 1 d μ + β A k , t | u | p 1 | H | d μ β 2 p r ( p r + 1 ) r p + 1 A k , t | u | p 1 | ϕ r ψ | r p + 1 d μ + β 2 p r ( p r + 1 ) r p + 1 A k , t | u | p 1 | r ϕ r 1 ϕ ( u ψ k + ) | r p + 1 d μ + β A k , t | u | p 1 | H | d μ β 2 p r ( p r + 1 ) r p + 1 ( A k , t | u | r d μ ) p 1 r ( A k , t | ψ | r d μ ) r p + 1 r + β 2 p r ( p r + 1 ) r p + 1 ( A k , t | u | r d μ ) p 1 r ( A k , t | r ϕ p 1 ϕ ( u ψ k + ) | r d μ ) r p + 1 r + β ( A k , t | u | r d μ ) p 1 r ( A k , t | H | r r p + 1 d μ ) r p + 1 r .
Let c 1 = 2 p - r ( p - r + 1 ) r - p + 1 , by (3.6) and Young's inequality
a b ε a 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 d μ β c 1 ε A k , t | u | r d μ + β c 1 c 2 ( ε , p ) A k , t | ψ | r d μ (1)  + β c 1 ε A k , t | u | r d μ + β c 1 c 2 ( ε , p ) A k , t | r ϕ r - 1 ϕ ( u - ψ k + ) | r d μ (2)  + β c A p ( w ) γ ( p - r ) ε A k , t | u | r d μ (3)  + β c A p ( w ) γ ( p - r ) c 2 ( ε , p ) Ω | ( v - u + T k ( u ) ) | r d μ , (4)  (5) 
where c is the constant given by Lemma 1. Since v - u + T k (u) = 0 on Ω\Ak,t, by the equality
v = ( u - T k ( u ) ) - ϕ r ( u - ψ k + ) - r ϕ r - 1 ϕ ( u - ψ k + ) ,
we obtain that
Ω | ( v - u + T k ( u ) ) | r d μ = A k , t | ( v - u ) | r d μ = A k , t | ϕ r ( u - ψ k + ) + r ϕ r - 1 ϕ ( u - ψ k + ) | r d μ 2 r - 1 A k , t | ( u - ψ k + ) | r d μ + 2 r - 1 r A k , t | ϕ ( u - ψ k + ) | r d μ 2 2 r - 2 A k , t | u | r d μ + 2 2 r - 2 A k , t | ψ | r d μ + r 2 2 r - 2 A k , t | u r | ( t - τ ) r d μ .
Finally, we obtain
A k , τ | u | r d μ β ( 2 c 1 + c A p ( w ) γ ( p r ) ) ε + β c A p ( w ) γ c 2 ( ε , p ) 2 2 r 2 ( p r ) α A k , t | u | r d μ + β c 1 c 2 ( ε , p ) + 2 2 r 2 β c A p ( w ) γ c 2 ( ε , p ) ( p r ) α A k , t | ψ | r d μ + r β c 1 c 2 ( ε , p ) + 2 2 r 1 β c A p ( w ) γ c 2 ( ε , p ) ( p r ) α A k , t | u | r ( t τ ) r d μ .
(3.8)
Now we want to eliminate the first term in the right-hand side containing u. Choosing ε and r1 such that
η = β ( 2 c 1 + c A p ( w ) γ ( p r ) ) ε + β c A p ( w ) γ c 2 ( ε , p ) 2 2 r 2 ( p r ) α < 1
and let ρ, R be arbitrarily fixed with R0ρ < RR1. Thus, from (3.8), we deduce that for every t and τ such that ρτ < tR, we have
A k , τ | u | r d μ η A k , t | u | r d μ + c 3 α A k , t | ψ | d μ + c 4 α ( t - τ ) r A k , t | u | r d μ ,
(3.9)
where
c 3 = β c 1 c 2 ( ε , p ) + 2 2 r - 2 β c A p ( w ) γ c 2 ( ε , p ) ( p - r )
and
c 4 = r β c 1 c 2 ( ε , p ) + r 2 2 r - 1 β c A p ( w ) γ c 2 ( ε , p ) ( p - r ) .
Applying Lemma 2 in (3.9), we conclude that
A k , ρ | u | r d μ c c 3 α A k , R | ψ | r d μ + c c 4 α ( R - ρ ) r A k , R | u | r d μ ,

where c is the constant given by Lemma 2. This ends the proof of the main theorem.

Declarations

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).

Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Hebei University
(2)
College of Mathematics and Computer Science, Hunan Normal University

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Copyright

© Hongya and Jinjing; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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 unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.