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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 [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 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 [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 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 [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.

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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).

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Correspondence to Gao Hongya.

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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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Keywords

  • Caccioppoli-type estimate
  • very weak solution
  • obstacle problem
  • Mucken-houpt weight
  • A-harmonic equation