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

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## 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

## 1 Introduction

Let w be a locally integrable non-negative function in Rn 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 Rn. 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 Rn.

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 . 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 Rn, where 2B is the ball with the same center as B and with radius twice that of B. Given a measurable subset E of Rn, we will denote by Lp(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 Lp (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 Rn, 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.

$⟨ A ( x , ξ ) , ξ ⟩ ≥αw ( x ) |ξ | p$,

2. 2.

$|A ( x , ξ ) |≤βw ( x ) |ξ | p - 1$,

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  in 1994, using the so-called reverse Hölder inequality. Gao and Tian  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 , and the regularity theory for very solutions of obstacle problems with w(x) ≡ 1 have been explored in . 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  which is a Hodge decomposition in weighted spaces.

Lemma 1 Let Ω be a regular domain of Rn 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  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 [, (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

(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

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.

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## 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 information

Correspondence to Gao Hongya.

### Competing interests

The authors declare that they have no competing interests.

### 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.

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Hongya, G., Jinjing, Q. A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight. J Inequal Appl 2011, 58 (2011). https://doi.org/10.1186/1029-242X-2011-58

• $A$-harmonic equation 