Open Access

Local boundedness results for very weak solutions of double obstacle problems

Journal of Inequalities and Applications20122012:43

https://doi.org/10.1186/1029-242X-2012-43

Received: 17 May 2011

Accepted: 27 February 2012

Published: 27 February 2012

Abstract

This article mainly concerns double obstacle problems for second order divergence type elliptic equation divA(x, u, u) = divf(x). We give local boundedness for very weak solutions of double obstacle problems.

Keywords

double obstacle problemslocal boundednesselliptic equation

1 Introduction

Let Ω be a bounded open set of R n , n 2. We consider the second order divergence type elliptic equation (also called A-harmonic equation or Leray-Lions equation)
div A ( x , u ( x ) , u ( x ) ) = div f ( x ) .
(1.1)
where A : Ω × R × R n R n is a Carathéodory function satisfying the coercivity and growth conditions: for almost all x Ω, all u R, and ξ R n ,
  1. (i)

    A(x, u, ξ), ξ〉 ≥ α |ξ| p ,

     
  2. (ii)

    |A(x, u, ξ)| ≤ β1 |ξ|p-1+ β2 |u| m + h(x),

     

where α > 0, β1 and β2 are some nonnegative constants, 1 < p < n, p - 1 m n ( p - 1 ) n - r and h ( x ) L loc s / ( p - 1 ) ( Ω ) , f ( x ) L loc s / ( p - 1 ) ( Ω ) n for some s > r.

Suppose that ψ1, ψ2 are any functions in Ω with values in R {±∞}, and that θ W1,r(Ω) with max{1, p - 1} < r ≤ p. Let
K ψ 1 , ψ 2 θ , r ( Ω ) = { v W 1 , r ( Ω ) : ψ 1 v ψ 2 , a . e . and v - θ W 0 1 , r ( Ω ) } .

The functions ψ1, ψ2 are two obstacles and θ determines the boundary values.

For any u , v K ψ 1 , ψ 2 θ , r ( Ω ) , we introduce the Hodge decomposition for | ( v - u ) | r - p ( v - u ) L r r - p + 1 , see [1]:
| ( v - u ) | r - p ( v - u ) = ϕ v , u + h v , u
(1.2)
where ϕ v , u W 0 1 , r r - p + 1 ( Ω ) , h v , u L r r - p + 1 ( Ω ) is a divergence-free vector field and the following estimates hold:
ϕ v , u r r - p + 1 c ( v - u ) r r - p + 1 ,
(1.3)
h v , u r r - p + 1 c ( p - r ) ( v - u ) r r - p + 1 ,
(1.4)

where c = c(n, p) is a constant depending only on n and p.

Definition 1.1. A very weak solution to the K ψ 1 , ψ 2 θ , r ( Ω ) -double obstacle problem for the A-harmonic Equation (1.1) is a function u K ψ 1 , ψ 2 θ , r ( Ω ) such that
Ω A ( x , u , u ) , | ( v - u ) | r - p ( v - u ) - h d x Ω f ( x ) , | ( v - u ) | r - p ( v - u ) - h d x ,
(1.5)

whenever v K ψ 1 , ψ 2 θ , r ( Ω ) .

The obstacle problem has a strong background, and has many applications in physics and engineering. 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. In [2], Gao et al. first considered the local boundedness for very weak solutions of obstacle problems to the A-harmonic equation in 2010. Precisely, the authors considered the local boundedness for very weak solutions of Kψ,θ(Ω)-obstacle problems to the A-harmonic equation div A(x, u(x)) = 0 with the obstacle function ψ 0, where operator A satisfies conditions 〈A(x, ξ), ξ α|ξ| p and |A(x, ξ)| ≤ β|ξ|p-1with A(x, 0) = 0. For the property of weak solutions of nonlinear elliptic equations, we refer the reader to [36].

In this article, we continue to consider the local boundedness property. Under some general conditions (i) and (ii) given above on the operator A, we obtain a local boundedness result for very weak solutions of K ψ 1 , ψ 2 θ , r -double obstacle problems to the A-harmonic Equation (1.1).

Theorem. Let operator A satisfies conditions (i) and (ii). Suppose that ψ 1 , ψ 2 W lo c 1 , ( Ω ) . Then a very weak solution u to the K ψ 1 , ψ 2 θ , r ( Ω ) -obstacle problem of (1.1) is locally bounded.

Remark. Since we have assumed that operator A satisfies the conditions (ii), in the proof of the theorem, we have to estimate the integral of some power of |u| by means of |u|. To deal with this difficulty, we will make use of the Sobolev inequality that was used in [4].

2 Preliminary knowledge and lemmas

We give some symbols and preliminary lemmas used in the proof. If x0 Ω and t > 0, then B t denotes the ball of radius t centered at x0. 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. Let t k (u) = min{u, k}.

Lemma 2.1. [7] Let f(τ) be a nonnegative bounded function defined for 0 R0 ≤ t ≤ R1. Suppose that for R0 ≤ τ < t ≤ R1 one has
f ( τ ) A ( t - τ ) - α + B + θ f ( t ) ,
where A, B, α, θ are nonnegative constants and θ < 1. Then there exists a constant c = c(α, θ), depending only on α and θ, such that for every ρ, R, R0 ≤ ρ < R ≤ R1, one has
f ( ρ ) c [ A ( R - ρ ) - α + B ] .
Definition 2.2. [8] A function u W lo c 1 , m ( Ω ) belongs to the class B(Ω, γ, m, k0), if for all k > k0, k0 > 0 and all B ρ = B ρ (x0), B ρ-ρσ = B ρ-ρσ (x0), B R = B R (x0), one has
A k , ρ - ρ σ + | u | m d x γ σ - m ρ - m A k , ρ + ( u - k ) m d x + | A k , ρ + | ,

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 + .

Lemma 2.3. [8] Suppose that u(x) is an arbitrary function belonging to the class B(Ω, γ, m, k0) and B R Ω. Then one has
max B R / 2 u ( x ) c ,

in which the constant c is determined only by γ, m, k0, R, ||u|| m .

3 Proof of theorem

Proof. Let u be a very weak solution to the K ψ 1 , ψ 2 θ , r ( Ω ) -obstacle problem for the A-harmonic Equation (1.1). Let B R 1 Ω and 0 < R1/ 2 τ < t R1 be arbitrarily fixed. Fix a cutoff function ϕ C 0 ( B R 1 ) , such that
supp ϕ B t , 0 ϕ 1 , ϕ 1 in B τ , | ϕ | 2 ( t - τ ) - 1 .
(3.1)
If ψ2 is an arbitrary function in Ω with values in R {+∞}, consider the function
v = u - ϕ r ( u - ψ k ) ,
(3.2)
where
ψ k = min { max { ψ 1 , t k ( u ) } , ψ 2 } , t k ( u ) = min { u , k } , k 0 .
It is easy to see ψ1ψ k ψ2. Now, v K ψ 1 , ψ 2 θ , r ( Ω ) ; indeed, since u K ψ 1 , ψ 2 θ , r ( Ω ) and ϕ C 0 ( Ω ) , then
v - θ = u - θ - ϕ r ( u - ψ k ) W 0 1 , r ( Ω ) , v - ψ 1 = u - ψ 1 - ϕ r ( u - ψ k ) ( 1 - ϕ r ) ( u - ψ 1 ) 0 a .e .in Ω , v - ψ 2 = u - ψ 2 - ϕ r ( u - ψ k ) ( 1 - ϕ r ) ( u - ψ 2 ) 0 a .e .in Ω .
(3.3)
For any fixed k > 0, let
v 0 = u , if u k , v , if u > k .
It is easy to see that v 0 K ψ 1 , ψ 2 θ , r ( Ω ) . Then by Definition 1.1 we have
Ω A ( x , u , u ) , | ( v 0 - u ) | r - p ( v 0 - u ) - h ̃ v , u d x Ω f ( x ) , | ( v 0 - u ) | r - p ( v 0 - u ) - h ̃ v , u d x .
(3.4)
If uk, then h ̃ v , u = 0 , ϕ ̃ v , u = 0 ; If u > k, then h ̃ v , u = h v , u , ϕ ̃ v , u = ϕ v , u . It's derived from the uniqueness of Hodge decomposition. This means that
A k , t + A ( x , u , u ) , h v , u d x + A k , t + f ( x ) , | ( v 0 - u ) | r - p ( v 0 - u ) - h ̃ v , u d x Ω A ( x , u , u ) , h v , u d x + Ω f ( x ) , | ( v 0 - u ) | r - p ( v 0 - u ) - h ̃ v , u d x Ω A ( x , u , u ) , | ( v 0 - u ) | r - p ( v 0 - u ) d x = Ω { u k } + Ω { u > k } A ( x , u , u ) , | ( v 0 - u ) | r - p ( v 0 - u ) d x = Ω { u > k } A ( x , u , u ) , | ( v 0 - u ) | r - p ( v 0 - u ) d x = A k , t + A ( x , u , u ) , | ( v - u ) | r - p ( v - u ) d x .
(3.5)
Let
E ( v , u ) = | ϕ r u | r - p ϕ r u + | ( v - u ) | r - p ( v - u ) .
(3.6)
By an elementary inequality [[9], P. 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 ,
(3.7)
v = u - ϕ r ( u - ψ k ) - r ϕ r - 1 ϕ ( u - ψ k ) ,
(3.8)
one can derive that
| E ( v , u ) | 2 p - r p - r + 1 r - p + 1 | ϕ r ψ k - r ϕ r - 1 ϕ ( u - ψ k ) | r - p + 1 .
(3.9)
We get from the definition of E(v, u) and (3.5) that
A k , t + A ( x , u , u ) , | ϕ r u | r - p ϕ r u d x = A k , t + A ( x , u , u ) , E ( v , u ) d x - A k , t + A ( x , u , u ) , | ( v - u ) | r - p ( v - u ) d x A k , t + A ( x , u , u ) , E ( v , u ) d x - A k , t + A ( x , u , u ) , h v , u d x - A k , t + f ( x ) , E ( v , u ) d x + A k , t + f ( x ) , | ϕ r u | r - p ϕ r u d x + A k , t + f ( x ) , h v , u d x = I 1 + I 2 + I 3 + I 4 + I 5 .
(3.10)
We now estimate the left-hand side and the right-hand side of (3.10), respectively. Firstly,
A k , t + A ( x , u , u ) , | ϕ r u | r - p ϕ r u d x A k , τ + A ( x , u , u ) , | u | r - p u d x α A k , τ + | u | r d x ,
(3.11)
here we have used condition (i). Secondly, by condition (ii) and (3.9),
| I 1 | = A k , t A ( x , u , u ) , E ( v , u ) d x A k , t + [ β 1 | u | p - 1 + β 2 | u | m + h 1 ] | E ( v , u ) | d x 2 p - r p - r + 1 r - p + 1 A k , t + [ β 1 | u | p - 1 + β 2 | u | m + h 1 ] | ϕ r ψ k - r ϕ r - 1 ϕ ( u - ψ k ) | r - p + 1 = C 1 β 1 A k , t + | u | p - 1 | ϕ r ψ k | r - p + 1 + C 1 β 1 A k , t + | u | p - 1 | r ϕ r - 1 ϕ ( u - ψ k ) | r - p + 1 + C 1 β 2 A k , t + | u | m | ϕ r ψ k | r - p + 1 + C 1 β 2 A k , t + | u | m | r ϕ r - 1 ϕ ( u - ψ k ) | r - p + 1 + C 1 A k , t + | h | ϕ r ψ k r - p + 1 + C 1 A k , t + | h | r ϕ r - 1 ϕ ( u - ψ k ) r - p + 1 = I 11 + I 12 + I 13 + I 14 + I 15 + I 16 ,
(3.12)
where C 1 = 2 p - r p - r + 1 r - p + 1 . By Young's inequality
a b ε a p + C ( ε , p ) b p valid for  a , b 0 , ε > 0 and  p > 1 ,
and p - 1 r + r - p + 1 r = 1 , we have the estimates
| I 11 | C 1 β 1 ε A k , t + | u | r d x + C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x ,
(3.13)
| I 12 | C 1 β 2 ε A k , t + | u | r d x + C ( ε , p ) A k , t + | r ϕ r - 1 ϕ ( u - ψ k ) | r d x ,
(3.14)
| I 13 | C 1 ε A k , t + | u | m r p - 1 d x + C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x ,
(3.15)

where we have used |ψ k | ≤ |ψ1| + |ψ2| in A k , t + .

We observe now that, if w W1,p(B t ) and | suppw| ≤ 1/ 2|B t |, then we have the Sobolev inequality (see also [10]),
B t | w | p * d x p / p * c 1 ( n , p ) B t | w | p d x .
(3.16)
Set
g k ( u ) = u , if u k , 0 , if u > k .
Since p - 1 m n ( p - 1 ) n - r by assumption, then r m r p - 1 r * . (3.16) implies
A k , t + | u | m r p - 1 d x = B t | u - g k ( u ) | m r p - 1 d x | | u - g k ( u ) | | r * m r p - 1 - r | B t | 1 - m r ( p - 1 ) r * B t | u - g k ( u ) | r * d x r / r * c 1 ( n , p ) | | u - g k ( u ) | | r * m r p - 1 - r | B t | 1 - m r ( p - 1 ) r * B t | ( u - g k ( u ) ) | r d x = c 1 ( n , p ) | | u - g k ( u ) | | r * m r p - 1 - r | B t | 1 - m r ( p - 1 ) r * A k , t + | u | r d x ,
(3.17)
provided that | supp( u g k ( u ) ) | B t | 1 / 2 | B t | . Since sup p ( u - g k ( u ) ) | B t A k , t + , then |supp | sup p ( u - g k ( u ) ) | B t | | A k , t + | . On the other hand, we have
| | u | | r * , B t r * = B t u r * d x A k , t + | u | r * d x k r * | A k , t + | .
Thus, there exists a constant k0 > 0, such that for all k ≥ k0, we have A k , t + 1 / 2 | B t | . We can also suppose that k0 such that
A k 0 , t u r * d x 1 .
For such values of k we then have inequality
A k , t + | u | m r p - 1 d x C 2 ( n , p ) | | u - g k ( u ) | | r * m r p - 1 - r | B t | 1 - m r ( p - 1 ) r * A k , t + | u | r d x C 2 ( n , p ) | | u - g k ( u ) | | r * m r p - 1 - r | Ω | 1 - m r ( p - 1 ) r * A k , t + | u | r d x C 2 ( n , p ) | Ω | 1 - m r ( p - 1 ) r * A k , t + | u | r d x = C 3 A k , t + | u | r d x ,
(3.18)
here C3 = C3(n, m, p, r, k0, | Ω|). We derive from (3.15) and (3.18) that
| I 13 | C 1 C 3 ε A k , t + | u | r d x + C 1 C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x .
(3.19)
| I 14 | C 1 C 3 ε A k , t + | u | r d x + C 1 C ( ε , p ) A k , t + | r ϕ r - 1 ϕ ( u - ψ k ) | r d x .
(3.20)
I15 and I16 can be estimated as follows:
| I 15 | C 1 ε A k , t + | h | r p - 1 d x + C 1 C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x .
(3.21)
| I 16 | C 1 ε A k , t + | h | r p - 1 d x + C 1 C ( ε , p ) A k , t + | r ϕ r - 1 ϕ ( u - ψ k ) | r d x .
(3.22)
In conclusion, we derive from (3.12)-(3.14), (3.19)-(3.22) that
| I 1 | ( C 1 β 1 ε + C 1 β 2 ε + 2 C 1 C 3 ε ) A k , t + | u | r d x + 2 C 1 ε A k , t + | h | r p - 1 d x + ( C 1 β 1 + 2 C 1 ) C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x + ( C 1 β 2 + 2 C 1 ) C ( ε , p ) A k , t + | r ϕ r - 1 ϕ ( u - ψ k ) | r d x .
(3.23)
By |ϕ| ≤ 2(t - τ)-1 and |u -ψ k | ≤ |u - k| a.e. in A k , t + , we have
| I 1 | C 4 ε A k , t + | u | r d x + 2 C 1 ε A k , t + | h | r p - 1 d x + C 5 C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x + C 6 C ( ε , p ) 2 r r ( t - τ ) r A k , t + | u - k | r d x .
(3.24)
We now estimate |I2|. By condition (ii),
| I 2 | = A k , t + A ( x , u , u ) , h v , u d x A k , t + [ β 1 | u | p - 1 + β 2 | u | m + h ] | h v , u | d x β 1 A k , t + | u | p - 1 | h v , u | d x + β 2 A k , t + | u | m | h v , u | d x + A k , t + | h | | h v , u | d x = I 21 + I 22 + I 23 .
(3.25)
By Young's inequality, Hölder's inequality and (1.4), I21 and I23 can be estimated as
| I 21 | β 1 A k , t + | u | r d x p - 1 r A k , t + | h v , u | r r - p + 1 d x r - p + 1 r β 1 C 7 ( p - r ) A k , t + | u | r d x p - 1 r A k , t + | ( v - u ) | r d x r - p + 1 r β 1 C 7 ( p - r ) ε A k , t + | u | r d x + β 1 C 7 ( p - r ) C ( ε , p ) A k , t + | ( v - u ) | r d x .
(3.26)
| I 23 | A k , t + | h | r p - 1 d x p - 1 r A k , t + | h v , u | r r - p + 1 d x r - p + 1 r C 7 ( p - r ) A k , t + | h | r p - 1 d x p - 1 r A k , t + | ( v - u ) | r d x r - p + 1 r C 7 ( p - r ) ε A k , t + | h | r p - 1 d x + C 7 ( p - r ) C ( ε , p ) A k , t + | ( v - u ) | r d x .
(3.27)
By (3.18), we know that if k ≥ k0, then
| I 22 | β 2 A k , t + | u | m r p - 1 d x p - 1 r A k , t + | h v , u | r r - p + 1 d x r - p + 1 r β 2 C 4 ( p - r ) A k , t + | u | m r p - 1 d x p - 1 r A k , t + | ( v - u ) | r d x r - p + 1 r β 2 C 7 ( p - r ) ε A k , t + | u | m r p - 1 d x + β 2 C 7 ( p - r ) C ( ε , p ) A k , t + | ( v - u ) | r d x β 2 C 7 ( p - r ) ε C 3 A k , t + | u | r d x + β 2 C 7 ( p - r ) C ( ε , p ) A k , t + | ( v - u ) | r d x .
(3.28)
Combining (3.25) with (3.26), (3.27), and (3.28), we obtain
| I 2 | ( β 1 C 4 + β 2 C 4 C 3 ) ( p - r ) ε A k , t + | u | r d x + C 4 ( p - r ) ε A k , t + | h | r p - 1 d x + ( β 1 C 4 + β 2 C 4 + C 4 ) ( p - r ) C ( ε , p ) A k , t + | ( v - u ) | r d x = C 8 ε A k , t + | u | r d x + C 9 A k , t + | h | r p - 1 d x + C 10 ( p - r ) C ( ε , p ) A k , t + | ( v - u ) | r d x .
(3.29)
We now estimate |I3|, |I4|, and |I5|.
| I 3 | = A k , t + f ( x ) , E v , u d x 2 p - r p - r + 1 r - p + 1 A k , t + | f ( x ) | ϕ r ψ k - r ϕ r - 1 ϕ ( u - ψ k ) r - p + 1 d x C 1 A k , t + | f ( x ) | | ϕ r ψ k | r - p + 1 d x + C 1 A k , t + | f ( x ) | r ϕ r - 1 ϕ ( u - ψ k ) r - p + 1 d x C 1 ε A k , t + | f ( x ) | r p - 1 d x + C 1 C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x + C 1 C ( ε , p ) A k , t + | r ϕ r - 1 ϕ ( u - ψ k ) | r d x C 1 ε A k , t + | f ( x ) | r p - 1 d x + C 1 C ( ε , p ) A k , t + | ψ 1 | r + | ψ 2 | r d x + C 1 C ( ε , p ) 2 r r ( t - τ ) r A k , t + | u - k | r d x .
(3.30)
| I 4 | = A k , t + f ( x ) , | ϕ r u | r - p ϕ r u d x A k , t + | f ( x ) | u r - p + 1 d x ε A k , t + | u | r d x + C ( ε , p ) A k , t + | f ( x ) | r p - 1 d x .
(3.31)
| I 5 | = A k , t + f ( x ) , h v , u d x A k , t + | f | r p - 1 d x p - 1 r A k , t + | h v , u | r r - p + 1 d x r - p + 1 r C 7 ( p - r ) A k , t + | f | r p - 1 d x p - 1 r A k , t + | ( v - u ) | r d x r - p + 1 r C 7 ( p - r ) ε A k , t + | f | r p - 1 d x + C 7 ( p - r ) C ( ε , p ) A k , t + | ( v - u ) | r d x .
(3.32)
By (3.8), we have
A k , t + | ( v - u ) | r d x A k , t + | u | r d x + A k , t + | ψ 1 | r + | ψ 2 | r d x + 2 r r ( t - τ ) r A k , t + | u - k | r d x .
(3.33)
Thus, the inequalities (3.10), (3.11), (3.24), and (3.29)-(3.33) imply that
A k , τ + | u | r d x 1 α { C 4 ε + C 8 ε + ε + ( C 10 + C 7 ) ( p - r ) C ( ε , p ) } A k , t + | u | r d x + 1 α { 2 C 1 ε + C 9 } A k , t + | h | r p - 1 d x + 1 α { C 1 ε + C ( ε , p ) + C 7 ( p - r ) ε } A k , t + | f | r p - 1 d x + 1 α { C 5 C ( ε , p ) + C 1 C ( ε , p ) + ( C 10 + C 7 ) ( p - r ) C ( ε , p ) } A k , t + | ψ 1 | r + | ψ 2 | r d x + 1 α { C 6 C ( ε , p ) + C 1 C ( ε , p ) + ( C 10 + C 7 ) ( p - r ) C ( ε , p ) } 2 r r ( t - τ ) r A k , t + | u - k | r d x .
(3.34)
Choosing ε and p-r small enough such that, the summation θ of the coefficients of the first term in the right-handside of (3.34) is smaller than 1. Let ρ, R be arbitrarily fixed with R1/ 2 ≤ ρ < R ≤ R1. Thus, from (3.34), we deduce that for every t and τ, such that R1/ 2 ≤ τ < t ≤ R1, we have
A k , τ + | u | r d x C 11 α A k , R + | ψ 1 | r + | ψ 2 | r + | h | r p - 1 + | f | r p - 1 d x + C 12 α ( t - τ ) r A k , R + | u - k | r d x + θ A k , t + | u | r d x ,
(3.35)
where C11,C12 are some constants depending only on n, p, r, m, k0, | Ω|, α, β1 and β2. Applying Lemma 2.1, we conclude that
A k , ρ + | u | r d x c C 11 α ( R - ρ ) r A k , R + | u - k | r d x + c C 12 α A k , R + | ψ 1 | r + | ψ 2 | r + | h 1 | r p - 1 + | h 2 | r p - 1 d x c C 11 α ( R - ρ ) r A k , R + | u - k | r d x + c C 12 C 13 α | A k , R + | ,
(3.36)
where c is the constant given by Lemma 2.1 and C 13 = | ψ 1 | r + | ψ 2 | r + | h | r p - 1 + | f | r p - 1 L ( Ω ) . Thus u belongs to the class B with γ = max{c2c7/α, c2c6c8/α} and m = r. Lemma 2.3 yields
max B R / 2 u ( x ) c .

If ψ1 is an arbitrary function in Ω with values in R {-∞}, noticing 2 ≤ -u ≤ -ψ1, we only use -u in place of u above.

These results together with the assumptions ψ1 ≤ u ≤ ψ2 and ψ1, ψ 2 W 1 oc 1 , ( Ω ) yield the desired result.

Declarations

Acknowledgements

The research was supported by the Natural Science Foundation of Hebei Province (A2010000910), Scientific Research Fund of Zhejiang Provincial Education Department(Y201016044) and Ningbo Natural Science Foundation(2011A610170).

Authors’ Affiliations

(1)
College of Science, Hebei United University
(2)
Department of Mathematics, Ningbo University

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