Integrability for Solutions of Anisotropic Obstacle Problems

(1)

Volume 2012, Article ID 549285,6pages doi:10.1155/2012/549285

Research Article

Integrability for Solutions of

Anisotropic Obstacle Problems

Hongya Gao, Yanjie Zhang, and Shuangli Li

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

Correspondence should be addressed to Hongya Gao,ghy@hbu.edu.cn

Received 30 March 2012; Revised 26 April 2012; Accepted 23 May 2012

Academic Editor: Martino Bardi

Copyrightq2012 Hongya Gao 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.

This paper deals with anisotropic obstacle problem for the A-harmonic equation

n

i1Diaix, Dux 0. An integrability result is given under suitable assumptions, which

show higher integrability of the boundary datum, and the obstacle force solutionsuhave higher integrability as well.

1. Introduction and Statement of Result

LetΩbe a bounded open subset of Rn. Forpi>1,i1,2, . . . , n, we denotepmmaxi1,2,...,npi

andpis the harmonic mean ofpi, that is,

1

p

1 n

n

i1 1

pi. 1.1

The anisotropic Sobolev spaceW1,piΩis defined by

W1,piΩ v∈W1,1Ω:Div∈LpiΩfor everyi1,2, . . . , n. 1.2

Let us consider solutionsu∈W1,piΩof the followingA-harmonic equation:

n

i1

(2)

whereD D1, D2, . . . , Dnis the gradient operator, and the Carath´eodory functionsaix, ξ:

Ω×Rn _→ _R,_i_1,_{2, . . . , n, satisfy}

|aix, z| ≤c2hx |zi|pi−1_, _1.4

for almost everyx∈Ω, for everyz∈Rn_{, and for any}_i_1,_{2, . . . , n, and there exists}_ν_∈_0,_∞

such that

ν n

i1

|zi−zi| pi _≤n

i1

aix, z−aix,zzi −zi, 1.5

for almost everyx∈Ω, for anyz,z∈Rn_{. The integrability condition for}_hx_≥_{0 in}_1.4_will

be given later.

Let ψ be any function in Ω with values in R∪ {±∞} and θ ∈ W1,piΩ, and we

introduce

Kpi_ψ,θΩ v∈W1,piΩ:v≥ψ, a.e.andv−θ∈W₀1,piΩ. 1.6

Note that

W₀1,piΩ v∈W₀1,1Ω:Div∈LpiΩfor everyi1,2, . . . , n. 1.7

The functionψis an obstacle andθdetermines the boundary values.

Definition 1.1. A solution to theKpi_ψ,θ-obstacle problem is a functionu∈ Kpi_ψ,θΩsuch that

Ω n

i1

aix, DuxDivx−Diuxdx≥0, 1.8

wheneverv∈ Kpi_ψ,θΩ.

Higher integrability property is important among the regularity theories of nonlinear

elliptic PDEs and systems, see the monograph1by Bensoussan and Frehse. Meyers and

Elcrat2first considered the higher integrability for weak solutions of1.3in 1975. Iwaniec

and Sbordone 3 obtained a regularity result for very weak solutions of theA-harmonic

equation1.3by using the celebrated Gehring’s Lemma. Global integrability for anisotropic

equation is contained in4. As far as higher integrability of∇uis concerned, in problems

with nonstandard growth a delicate interplay between the regularity with respect toxand

the growth with respect toξappears: see5. For a global boundedness result of anisotropic

variational problems, see6. For other related works, see7. We refer the readers to the

classical books by Ladyˇzenskaya and Ural’ceva8, Morrey9, Gilbarg and Trudinger10

and Giaquinta11for some details of isotropic cases.

In the present paper, we consider integrability for solutions of anisotropic obstacle

(3)

datum, and the obstacle force solutionsu, have higher integrability as well. The idea of this

paper comes from4, and the result can be considered as a generalization of4, Theorem 2.1.

Theorem 1.2. Letu ∈ Kpi_ψ,θΩbe a solution to the Kpi_ψ,θ obstacle problem andθ ∈ W1,qiΩ,

qi ∈pi,∞,i1,2, . . . , n, 0 ≤h∈Lqm_Ω_with_qm _max_i1,...,nqi_,_ψ _∈_−∞,_∞_{is such that}

θ∗max{ψ, θ} ∈θW01,qiΩ. Moreover,p < n. Then

u∈θ∗LtweakΩ, 1.9

where

t p

∗

1−bp∗/p pm/pm−1 > p

∗_, _1.10

andbis any number verifying

0< b≤ min

j1,...,n

1−pj

qj

1− 1

pj

,

b <pm−1

pm p

p∗.

1.11

Remark 1.3. Take the obstacle function ψ to be minus infinity in Theorem 1.2, and the

condition1.4replaced by

|aix, z| ≤c21|zi|pi−1 _1.2

for almost every x ∈ Ω, for every z ∈ Rn_{, and for any} _i _1,_{2, . . . , n, then we arrive at}

Theorem 2.1 in4.

2. Proof of the Main Theorem

Proof ofTheorem 1.2. Letu ∈ Kpi_ψ,θΩbe a solution to theKpi_ψ,θ-obstacle problem. Takeθ∗

max{ψ, θ} ∈θW₀1,qiΩ. Let us considerL∈0,∞and

v

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

θ∗−L, foru−θ∗<−L,

u, for −L≤u−θ∗≤L,

θ∗L, foru−θ∗> L.

2.1

Thenv ∈ Kpi_ψ,θΩ. Indeed, for the second and the third cases of the above definition forv,

(4)

impliesvθ∗−L≥ψ. Sinceuθ∗θon∂Ω, thenvuon∂Ω, this impliesvθon∂Ω. By

Definition 1.1, one has

0≤

{|u−θ∗|>L}

n

i1

aix, DuxDivx−Diuxdx

{|u−θ∗|>L}

n

i1

aix, DuxDiθ∗x−Diuxdx.

2.2

Monotonicity1.5allows us to write

ν n i1

{|u−θ∗|>L}

|Diux−Diθ∗x|pidx

≤

{|u−θ∗|>L}

n

i1

aix, Dux−aix, Dθ∗xDiux−Diθ∗xdx,

2.3

which together with2.2implies

ν n i1

{|u−θ∗|>L}|Diux−Diθ∗x|

pi_dx

≤ −

{|u−θ∗|>L}

n

i1

aix, Dθ∗Diux−Diθ∗xdx.

2.4

We now use anisotropic growth1.4and the H ¨older inequality in2.4, obtaining that

ν n i1

{|u−θ∗|>L}|Diu−Diθ∗|

pi dx

≤ −n

i1

{|u−θ∗|>L}aix, Dθ∗Diu−Diθ∗dx

≤c2

n

i1

{|u−θ∗|>L}

h|Diθ∗|pi−1|Diu−Diθ∗|dx

≤c2

n

i1

{|u−θ∗|>L}h|Diθ∗|

pi_dx

pi−1/pi

pi_dx

1/pi .

2.5

Lettibe such that

(5)

for everyi1, . . . , n;tiwill be chosen later. We use the H ¨older inequality as follows:

pi dx

_pi₋_1/pi

≤

{|u−θ∗|>L}

h|Diθ∗|tidx

_pi₋_1/ti

|{|u−θ∗|> L}|ti−pipi−1/tipi.

2.7

The following proof is similar to that of4, Theorem 2.1; we only list the necessary changes:

instead of4,3.14by

{|u−θ∗|>L}

h|Diθ∗|pidx

pi−1/pi

≤

ti dx

pi−1/ti

|{|u−θ∗|> L}|b

≤M|{|u−θ∗|> L}|b,

2.8

where

M max

j1,...,n

Ω

h|Djθ∗| tjdx

_pj₋_1/tj

<∞, 2.9

and instead of4,3.19we use anisotropic Sobolev Embedding Theorem forv−u,

Ω|v−u|

p∗_dx

1/p∗

≤c∗

_n

i1

Ω|Div−u|

pi_dx

1/pi1/n

≤c∗

⎡

⎣n

i1

pi_dx

1/pi⎤

⎦ 1/n

.

2.10

By|v−u| |u−θ∗| −L1{|u−θ∗|>L}, we obtain

{|u−θ∗|>L}|u−θ∗| −L

p∗

dx

1/p∗

Ω|v−u|

p∗

dx

1/p∗

. 2.11

(6)

Acknowledgment

The authors would like to thank the referee for valuable suggestions. Supported by NSFC

10971224and NSF of Hebei ProvinceA2011201011.

References

1 A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, vol. 151, Springer, Berlin, Germany, 2002.

2 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.

3 T. Iwaniec and C. Sbordone, “Weak minima of variational integrals,” Journal f ¨ur die Reine und

Angewandte Mathematik, vol. 454, pp. 143–161, 1994.

4 F. Leonetti and F. Siepe, “Integrability for solutions to some anisotropic elliptic equations,” Nonlinear

Analysis: Theory, Methods & Applications, vol. 75, pp. 2867–2873, 2012.

5 L. Esposito, F. Leonetti, and G. Mingione, “Sharp regularity for functionals withp, qgrowth,” Journal

of Diﬀerential Equations, vol. 204, no. 1, pp. 5–55, 2004.

6 B. Stroﬀolini, “Global boundedness of solutions of anisotropic variational problems,” Unione

Matematica Italiana, vol. 7, no. 5, pp. 345–352, 1991.

7 H. Y. Gao and and Q. H. Huang, “Local regularity for solutions of anisotropic obstacle problems,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 13, pp. 4761–4765, 2012.

8 O. A. Ladyˇzenskaya and N. N. Ural’ceva, Linear and Quasilinear Elliptic Equations, Academic Press, 1968.

9 C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, 1968.

10 D. Gilbarg and N. S. Trudinger, Elliptic Partial Diﬀerential Equations of Second Order, vol. 224, Springer,

Berlin, Germany, 1977.

11 M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105 of

(7)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

http://www.hindawi.com Volume 2014 International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

Journal of

Integrability for Solutions of Anisotropic Obstacle Problems

Research Article