Local Regularity and Local Boundedness Results for Very Weak Solutions of Obstacle Problems

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Journal of Inequalities and Applications Volume 2010, Article ID 878769,12pages doi:10.1155/2010/878769

Research Article

Local Regularity and Local Boundedness Results for Very Weak Solutions of Obstacle Problems

Gao Hongya,

^{1, 2}

Qiao Jinjing,

³

and Chu Yuming

⁴

1College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

2Hebei Provincial Center of Mathematics, Hebei Normal University, Shijiazhuang 050016, China

3College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China

4Faculty of Science, Huzhou Teachers College, Huzhou, Zhejiang 313000, China

Correspondence should be addressed to Gao Hongya,[email protected] Received 25 September 2009; Accepted 18 March 2010

Academic Editor: Yuming Xing

Copyrightq2010 Gao Hongya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Local regularity and local boundedness results for very weak solutions of obstacle problems of the A-harmonic equation divAx,∇ux 0 are obtained by using the theory of Hodge decomposition, where|Ax, ξ| ≈ |ξ|^p−1.

1. Introduction and Statement of Results

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 in1.5 and 1.6are satisfied; see1. A Lipschitz domain, for example, is a regular domain. We consider the second-order divergence type elliptic equationalso calledA-harmonic equation or Leray-Lions equation:

divAx,∇ux 0, 1.1

whereAx, ξ:Ω×Rⁿ → Rⁿis a Carath´eodory function satisfying the following conditions:

aAx, ξ, ξ ≥α|ξ|^p, b|Ax, ξ| ≤β|ξ|^p−1, cAx,0 0,

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wherep >1 and 0< α≤β <∞. The prototype of1.1is thep-harmonic equation:

div

|∇u|^p−2∇u

0. 1.2

Suppose thatψis an arbitrary function inΩwith values in R∪ {±∞}, andθ∈W^1,rΩwith max{1, p−1}< r ≤p. Let

K^r_ψ,θΩ

v∈W^1,rΩ:v≥ψ a.e., andv−θ∈W₀^1,rΩ

. 1.3

The functionψis an obstacle andθdetermines the boundary values.

For anyu, v∈ K^r_ψ,θΩ, we introduce the Hodge decomposition for|∇v−u|^r−p∇v− u∈L^r/r−p1Ω, see1:

|∇v−u|^r−p∇v−u ∇φv,uhv,u, 1.4

whereφv,u ∈ W₀^1,r/r−p1Ωand hv,u ∈ L^r/r−p1Ω,Rⁿare a divergence-free vector field, and the following estimates hold:

∇φv,u

r/r−p1≤c1∇v−u^r−p1r , 1.5

hv,u_r/r−p1≤c1

p−r

∇v−u^r−p1r , 1.6

wherec1c1n, pis some constant depending only onnandp.

Definition 1.1see2. A very weak solution to theK^r_ψ,θ-obstacle problem is a functionu∈ K^r_ψ,θΩsuch that

Ω

Ax,∇u,|∇v−u|^r−p∇v−u dx≥

ΩAx,∇u, hv,udx, 1.7

wheneverv∈ K^r_ψ,θΩ.

Remark 1.2. If r p in Definition 1.1, then hv,u 0 by the uniqueness of the Hodge decomposition1.4, and1.7becomes

ΩAx,∇u,∇v−udx≥0. 1.8 This is the classical definition forK^p_ψ,θ-obstacle problem; see3for some details of solutions ofK^p_ψ,θ-obstacle problem.

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This paper deals with local regularity and local boundedness for very weak solutions of obstacle problems. Local regularity and local boundedness properties are important among the regularity theories of nonlinear elliptic systems; see the recent monograph4 by Bensoussan and Frehse. Meyers and Elcrat5first considered the higher integrability for weak solutions of1.1in 1975; see also6. Iwaniec and Sbordone1obtained the regularity result for very weak solutions of the A-harmonic1.1 by using the celebrated Gehring’s Lemma. The local and global higher integrability of the derivatives in obstacle problem was first considered by Li and Martio7in 1994 by using the so-called reverse H ¨older inequality.

Gao et al.2gave the definition for very weak solutions of obstacle problem ofA-harmonic 1.1and obtained the local and global higher integrability results. The local regularity results for minima of functionals and solutions of elliptic equations have been obtained in8. For some new results related to A-harmonic equation, we refer the reader to 9–11. Gao and Tian 12 gave the local regularity result for weak solutions of obstacle problem with the obstacle functionψ ≥0. Li and Gao13generalized the result of12by obtaining the local integrability result for very weak solutions of obstacle problem. The main result of13is the following proposition.

Proposition 1.3. There existsr1with max{1, p−1}< r1< p, such that any very weak solutionuto theK^r_ψ,θ-obstacle problem belongs toL^s_loc^∗Ω,s^∗1/1/s−1/n, provided that 0≤ψ ∈W_loc^1,sΩ, r < s < n, andr1< r <min{p, n}.

Notice that in the above proposition we have restricted ourselves to the caser < n, because whenr ≥n, every function inW_loc^1,rΩis trivially inL^t_locΩfor everyt > 1 by the classical Sobolev imbedding theorem.

In the first part of this paper, we continue to consider the local regularity theory for very weak solutions of obstacle problem by showing that the condition ψ ≥ 0 in Proposition 1.3is not necessary.

Theorem 1.4. There exists r1 with max{1, p−1} < r1 < p, such that any very weak solution uto the K^r_ψ,θ-obstacle problem belongs toL^s_loc^∗Ω, provided that ψ ∈ W_loc^1,sΩ,r < s < n, and r1< r <min{p, n}.

As a corollary of the above theorem, ifr p, that is, if we consider weak solutions of K^p_ψ,θ-obstacle problem, then we have the following local regularity result.

Corollary 1.5. Suppose thatψ ∈ W_loc^1,sΩ, 1< p < s < n. Then a solutionuto theK^p_ψ,θ-obstacle problem belongs toL^s_loc^∗ Ω.

We omit the proof of this corollary. This corollary shows that the conditionψ ≥0 in the main result of12is not necessary.

The second part of this paper considers local boundedness for very weak solutions of K^r_ψ,θ-obstacle problem. 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. For the local boundedness results of weak solutions of nonlinear elliptic equations, we refer the reader to4. In this paper we consider very weak solutions and show that if the obstacle function isψ ∈ W_loc^1,∞Ω, then a very weak solutionuto the K_ψ,θ^r -obstacle problem is locally bounded.

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Theorem 1.6. There exists r1 with max{1, p−1} < r1 < p, such that for any r withr1 < r <

min{p, n}and any ψ ∈ W_loc^1,∞Ω, a very weak solution u to theK^r_ψ,θ-obstacle problem is locally bounded.

Remark 1.7. As far as we are aware, Theorem 1.6 is the first result concerning local boundedness for very weak solutions of obstacle problems.

In the remaining part of this section, we give some symbols and preliminary lemmas used in the proof of the main results. Ifx0 ∈Ωandt >0, thenBtdenotes the ball of radius tcentered atx0. For a functionuxandk >0, letAk {x∈Ω:|ux|> k},A_k {x∈Ω: ux > k},Ak,t Ak∩Bt,A_k,t A_k∩Bt. Moreover ifs < n,s^∗is always the real number satisfying 1/s^∗1/s−1/n. LetTkube the usual truncation ofuat levelk >0, that is,

Tku max{−k,min{k, u}}. 1.9

Lettku min{u, k}.

We recall two lammas which will be used in the proof ofTheorem 1.4.

Lemma 1.8see8. Letu∈W_loc^1,rΩ,ϕ0∈L^q_locΩ, where 1< r < nandqsatisfies

1< q < n

r. 1.10

Assume that the following integral estimate holds:

Ak,t

|∇u|^rdx≤c0 Ak,t

ϕ0dx t−τ^−α

Ak,t

|u|^rdx

, 1.11

for everyk ∈ N andR0 ≤ τ < t ≤ R1, wherec0 is a real positive constant that depends only on N, q, r, R0, R1,|Ω|andαis a real positive constant. Thenu∈L^qr_loc ^∗Ω.

Lemma 1.9see14. Letfτbe a nonnegative bounded function defined for 0 ≤R0 ≤ t ≤R1. Suppose that forR0≤τ < t≤R1one has

fτ≤At−τ^−αBθft, 1.12

where A, B, α, θ are nonnegative constants andθ < 1. Then there exists a constantc2 c2α, θ, depending only onαandθ, such that for everyρ, R, R0≤ρ < R≤R1one has

f ρ

≤c2

A

R−ρ_−α B

. 1.13

We need the following definition.

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Definition 1.10see15. A functionux∈W_loc^1,mΩbelongs to the classBΩ, γ, m, k0, if for allk > k0,k0>0 and allBρBρx0,B_ρ−ρσB_ρ−ρσx0,BR BRx0, one has

A_k,ρ−ρσ

|∇u|^mdx≤γ

σ^−mρ^−m

A_k,ρ

u−k^mdxA_k,ρ

, 1.14

forR/2≤ρ−ρσ < ρ < R,m < n, where|A_k,ρ|is then-dimensional Lebesgue measure of the setA_k,ρ.

We recall a lemma from15which will be used in the proof ofTheorem 1.6.

Lemma 1.11 see 15. Suppose that ux is an arbitrary function belonging to the class BΩ, γ, m, k0andBR⊂⊂Ω. Then one has

maxBR/2

ux≤c, 1.15

in which the constantcis determined only by the quantitiesγ, m, k0, R,∇u_m₁.

2. Local Regularity

Proof ofTheorem 1.4. Let u be a very weak solution to the K^r_ψ,θ-obstacle problem. By Lemma 1.8, it is suﬃcient to prove that u satisfies the inequality 1.11 with α r. Let BR1⊂⊂Ωand 0< R0≤τ < t≤R1be arbitrarily fixed. Fix a cut-oﬀfunctionφ∈C₀^∞BR1such that

suppφ⊂Bt, 0≤φ≤1, φ1 inBτ,∇φ≤2t−τ⁻¹. 2.1

Consider the function

vu−Tku−φ^r u−ψk

, 2.2

whereTkuis the usual truncation ofuat levelk≥0 defined in1.9andψkmax{ψ, Tku}.

Nowv∈ K^r_ψ−T

ku,θ−TkuΩ; indeed, sinceu∈ K^r_ψ,θΩandφ∈C^∞₀Ω, then

v−θ−Tku u−θ−φ^r u−ψk

∈W₀^1,rΩ, v−

ψ−Tku

u−ψ−φ^r u−ψk

≥

1−φ^r u−ψ

≥0, 2.3

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a.e. inΩ. Let

Ev, u φ^r∇u^r−pφ^r∇u|∇v−uTku|^r−p∇v−uTku, 2.4

By an elementary inequality16, Page 271,4.1, |X|^−εX− |Y|^−εY≤2^ε1ε

1−ε|X−Y|^1−ε, X, Y∈Rⁿ, 0≤ε <1,

∇v∇u−Tku−φ^r∇ u−ψk

−rφ^r−1∇φ u−ψk

,

2.5

one can derive that

|Ev, u| ≤2^p−rp−r1 r−p1

φ^r∇ψk−rφ^r−1∇φ

u−ψk^r^−p1. 2.6 We get from the definition ofEv, uthat

Ak,t

Ax,∇u,φ^r∇u^r−pφ^r∇u dx

Ak,t

Ax,∇u, Ev, udx

−

Ak,t

Ax,∇u,|∇v−uTku|^r−p∇v−uTku dx

Ak,t

Ax,∇u, Ev, udx

−

Ak,t

Ax,∇u,|∇v−u|^r−p∇v−u dx.

2.7

Now we estimate the left-hand side of2.7. By conditionawe have

Ak,t

Ax,∇u,φ^r∇u^r−pφ^r∇ud≥

Ak,τ

Ax,∇u,|∇u|^r−p∇udx≥α

Ak,τ

|∇u|^rdx. 2.8

Sinceu−Tku, v∈ K^r_ψ−T_k_u,θ−T_k_uΩ, then using the Hodge decomposition1.4, we get

|∇v−uTku|^r−p∇v−uTku ∇φh, 2.9

and by1.6we have

h_r/r−p1≤c1

p−r

∇v−uTku^r−p1r . 2.10

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Thus we derive, byDefinition 1.1, that

Ω

Ax,∇u−Tku,|∇v−uTku|^r−p∇v−uTku dx

≥ ΩAx,∇u−Tku, hdx.

2.11

This means, by conditionc, that

Ak,t

Ax,∇u,|∇v−u|^r−p∇v−udx≥

Ak,t

Ax,∇u, hdx. 2.12

Combining the inequalities 2.7, 2.8, and 2.12, and using H ¨older’s inequality and conditionb, we obtain

α

Ak,τ

|∇u|^rdx≤

Ak,t

Ax,∇u, Ev, udx−

Ak,t

Ax,∇u, hdx

≤β2^p−r

p−r1

r−p1 _A_k,t|∇u|^p−1φ^r∇ψk−rφ^r−1∇φ

u−ψk^r−p1dx β

Ak,t

|∇u|^p−1|h|dx

≤β2^p−r

p−r1

r−p1 _A_k,t|∇u|^p−1φ^r∇ψk^r−p1dx β2^p−r

p−r1

r−p1 _A_k,t|∇u|^p−1rφ^r−1∇φ

u−ψk^r−p1dx

β

Ak,t

|∇u|^p−1|h|dx

≤β2^p−r

p−r1

r−p1 _A_k,t|∇u|^rdx _p−1/r

Ak,t

∇ψk^rdx

_r−p1/r

β2^p−r

p−r1

r−p1 _A_k,t|∇u|^rdx

_p−1/r

×

Ak,t

rφ^r−1∇φ

u−ψk^rdx

_r−p1/r

β

Ak,t

|∇u|^rdx

_p−1/r

Ak,t

|h|^r/r−p1dx

_r−p1/r .

2.13

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Denotec3c3p, r 2^p−rp−r1/r−p1. It is obvious that ifris suﬃciently close top, thenc3p, r≤2. By2.10and Young’s inequality

ab≤εa^pc4

ε, p b^p, 1

p 1

p 1, a, b≥0, ε≥0, p≥1, 2.14

we can derive that

α

Ak,τ

|∇u|^rdx≤βc3

p, r ε

Ak,t

|∇u|^rdxβc3

p, r c4

ε, p

Ak,t

∇ψk^rdx βc3

p, r ε

Ak,t

|∇u|^rdxβc3

p, r c4

ε, p

Ak,t

rφ^r−1∇φ

u−ψk^rdx βc1ε

p−r

Ak,t

|∇u|^rdxβc1c4

ε, p p−r

Ω|∇v−uTku|^rdx

≤βε 2c3

p, r c1

p−r

Ak,t

|∇u|^rdxβc3

p, r c4

ε, p

Ak,t

∇ψk^rdx βc3

p, r c4

ε, p

Ak,t

rφ^r−1∇φ

u−ψk^rdx βc1c4

ε, p p−r

Ω|∇v−uTku|^rdx.

2.15

By the equality

∇v∇u−Tku−φ^r∇ u−ψk

−rφ^r−1∇φ u−ψk

, 2.16

andv−uTku 0 forx∈Ω\Ak,t, then we have

Ω|∇v−uTku|^rdx

Ak,t

φ^r∇ u−ψk

rφ^r−1∇φ

u−ψk^rdx

≤2^r−1

Ak,t

|∇u|^rdx

Ak,t

∇ψk^rdx

Ak,t

rφ^r−1∇φ

u−ψk^rdx

. 2.17

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Finally we obtain that

Ak,τ

|∇u|^rdx≤ βε2c3

p, r c1

p−r

2^r−1βc1c4

ε, p p−r

α _A_k,t|∇u|^rdx

βc3

p, r c4

ε, p

2^r−1βc1c4

ε, p p−r

α _A_k,t∇ψk^rdx

βc3

p, r c4

ε, p

2^r−1βc1c4

ε, p p−r

α _A_k,t

rφ^r−1∇φ

u−ψk^rdx

≤ βε2c3

p, r c1

p−r

2^p−1βc1c4

ε, p p−r

α _A_k,t|∇u|^rdx

βc3

p, r c4

ε, p

2^p−1βc1c4

ε, p p−r

α _A_k,t∇ψ^rdx

βc3

p, r c4

ε, p

2^p−1βc1c4

ε, p p−r α

2^pp

t−τ^r _A_k,t|u|^rdx.

2.18

The last inequality holds since|u−ψk| ≤ |u|a.e. inAk,t. Now we want to eliminate the first term in the right-hand side containing∇u. Chooseεsmall enough andrsuﬃciently close to psuch that

θ βε2c3

p, r c1

p−r

2^p−1βc1c4

ε, p p−r

α <1, 2.19

and letρ, Rbe arbitrarily fixed withR0 ≤ρ < R ≤R1. Thus, from2.18, we deduce that for everyτandtsuch thatρ≤τ < t≤R, we have

Ak,τ

|∇u|^rdx≤θ

Ak,t

|∇u|^rdxc5

α _A_k,R∇ψ^rdx c6

αt−τ^r Ak,R

|u|^rdx, 2.20

wherec5 βc3p, rc4ε, p 2^p−1βc1c4ε, pp−rwith εandr fixed to satisfy2.19, and c62^ppc5. ApplyingLemma 1.9in2.20we conclude that

Ak,ρ

|∇u|^rdx≤ c2c5

α _A_k,R∇ψ^rdx c2c6

α R−ρr

Ak,R

|u|^rdx, 2.21

wherec2is the constant given byLemma 1.9. Thususatisfies inequality1.11withϕ0|∇ψ|^r andαr.Theorem 1.4follows fromLemma 1.8.

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3. Local Boundedness

Proof ofTheorem 1.6. Letube a very weak solution to theK^r_ψ,θ-obstacle problem. LetBR1⊂⊂Ω andR1/2≤τ < t≤R1be arbitrarily fixed. Fix a cut-oﬀfunctionφ∈C^∞₀ BR1such that

suppφ⊂Bt, 0≤φ≤1, φ1 inBτ,∇φ≤2t−τ⁻¹. 3.1

Consider the function

vu−tku−φ^r

u−max

ψ, tku

, 3.2

wheretku min{u, k}. Nowv∈ K^r_ψ−t

ku,θ−tku; indeed, sinceu∈ K^r_ψ,θΩandφ∈C^∞₀ Ω, then

v−θ−tku u−θ−φ^r

u−max

ψ, tku

∈W₀^1,rΩ, v−

ψ−tku

u−ψ−φ^r

u−max

ψ, tku

≥

1−φ^r u−ψ

≥0 3.3 a.e. inΩ.

As in the proof ofTheorem 1.4, we obtain

A_k,τ

|∇u|^rdx≤ βε2c3

p, r c1

p−r

2^r−1βc1c4

ε, p p−r

α _A

k,t

|∇u|^rdx βc3c4

ε, p

2^r−1βc1c4

ε, p p−r

α _A

k,t

∇max

ψ, tku^rdx βc3

p, r c4

ε, p

2^r−1βc1c4

ε, p p−r α

×

A_k,t

rφ^r−1∇φu−max{ψ, tku}^rdx

≤ βε2c3

p, r c1

p−r

2^r−1βc1c4

ε, p p−r

α _A

k,t

|∇u|^rdx βc3c4

ε, p

2^r−1βc1c4

ε, p p−r

α _A

k,t

∇ψ^rdx βc3

p, r c4

ε, p

2^r−1βc1c4

ε, p p−r α

2^pp t−τ^r A_k,t

|u−k|^rdx.

3.4

Choose ε small enough and r1 suﬃciently close to p such that 2.19 holds. Let ρ, R be arbitrarily fixed withR1/2 ≤ ρ < R ≤ R1. Thus from3.4we deduce that for everyτ and tsuch thatR1/2≤τ < t≤R1, we have

A_k,τ

|∇u|^rdx≤θ

A_k,t

|∇u|^rdxc5

α _A

k,R

∇ψ^rdx c6

αt−τ^r _A

k,R

|u−k|^rdx. 3.5

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ApplyingLemma 1.9, we conclude that

A_k,ρ

|∇u|^rdx≤ c2c6

α R−ρ_r

A_k,R

|u−k|^rdxc2c5

α A_k,R

∇ψ^rdx

≤ c2c6

α R−ρ_r

A_k,R

|u−k|^rdxc2c5c7

α

A_k,R,

3.6

wherec2is the constant given byLemma 1.9andc7∇ψ^p_L∞Ω. Thusubelongs to the class Bwithγmax{c2c6/α, c2c5c7/α}andmr.Lemma 1.11yields

maxBR/2

ux≤c. 3.7

This result together with the assumptionsu≥ψandψ ∈W_loc^1,∞Ωyields the desired result.

Acknowledgments

The authors would like to thank the referee of this paper for helpful comments upon which this paper was revised. The first author is supported by NSFC10971224and NSF of Hebei Province07M003. The third author is supported by NSF of Zhejiang provinceY607128 and NSFC10771195.

References

1 T. Iwaniec and C. Sbordone, “Weak minima of variational integrals,” Journal f ¨ur die reine und angewandte Mathematik, vol. 454, pp. 143–161, 1994.

2 H. Y. Gao, M. Wang, and H. L. Zhao, “Very weak solutions for obstacle problems of theA-harmonic equation,” Journal of Mathematical Research and Exposition, vol. 24, no. 1, pp. 159–167, 2004Chinese.

3 J. Heinonen, T. Kilpel¨ainen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford, UK, 1993.

4 A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, vol. 151 of Applied Mathematical Sciences, Springer, Berlin, Germany, 2002.

5 N. G. Meyers and A. Elcrat, “Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions,” Duke Mathematical Journal, vol. 42, pp. 121–136, 1975.

6 E. W. Strdulinsky, “Higher integrability from reverse H ¨older inequalities,” Indiana University Mathematics Journal, vol. 29, pp. 408–413, 1980.

7 G. B. Li and O. Martio, “Local and global integrability of gradients in obstacle problems,” Annales Academiae Scientiarum Fennicae. Series A, vol. 19, no. 1, pp. 25–34, 1994.

8 D. Giachetti and M. M. Porzio, “Local regularity results for minima of functionals of the calculus of variation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 39, no. 4, pp. 463–482, 2000.

9 Y. Xing and S. Ding, “Inequalities for Green’s operator with Lipschitz and BMO norms,” Computers &

Mathematics with Applications, vol. 58, no. 2, pp. 273–280, 2009.

10 S. Ding, “Lipschitz and BMO norm inequalities for operators,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 71, no. 12, pp. e2350–e2357, 2009.

11 H. Gao, J. Qiao, Y. Wang, and Y. Chu, “Local regularity results for minima of anisotropic functionals and solutions of anisotropic equations,” Journal of Inequalities and Applications, vol. 2008, Article ID 835736, 11 pages, 2008.

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12 H. Gao and H. Tian, “Local regularity result for solutions of obstacle problems,” Acta Mathematica Scientia B, vol. 24, no. 1, pp. 71–74, 2004.

13 J. Li and H. Gao, “Local regularity result for very weak solutions of obstacle problems,” Radovi Matematiˇcki, vol. 12, no. 1, pp. 19–26, 2003.

14 M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1983.

15 M. C. Hong, “Some remarks on the minimizers of variational integrals with nonstandard growth conditions,” Bollettino dell’Unione Matematica Italiana, vol. 6, no. 1, pp. 91–101, 1992.

16 T. Iwaniec, L. Migliaccio, L. Nania, and C. Sbordone, “Integrability and removability results for quasiregular mappings in high dimensions,” Mathematica Scandinavica, vol. 75, no. 2, pp. 263–279, 1994.