Document 10455063

advertisement
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 549285, 6 pages
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, [email protected]
Received 30 March 2012; Revised 26 April 2012; Accepted 23 May 2012
Academic Editor: Martino Bardi
Copyright q 2012 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.
paper deals with anisotropic obstacle problem for the A-harmonic equation
Di ai x, Dux 0. An integrability result is given under suitable assumptions, which
show higher integrability of the boundary datum, and the obstacle force solutions u have higher
integrability as well.
This
n
i1
1. Introduction and Statement of Result
Let Ω be a bounded open subset of Rn . For pi > 1, i 1, 2, . . . , n, we denote pm maxi1,2,...,n pi
and p is the harmonic mean of pi , that is,
n
1
1 1
.
p n i1 pi
1.1
The anisotropic Sobolev space W 1,pi Ω is defined by
W 1,pi Ω v ∈ W 1,1 Ω : Di v ∈ Lpi Ω for every i 1, 2, . . . , n .
1.2
Let us consider solutions u ∈ W 1,pi Ω of the following A-harmonic equation:
n
Di ai x, Dux 0,
i1
1.3
2
International Journal of Mathematics and Mathematical Sciences
where D D1 , D2 , . . . , Dn is the gradient operator, and the Carathéodory functions ai x, ξ :
Ω × Rn → R, i 1, 2, . . . , n, satisfy
|ai x, z| ≤ c2 hx |zi |pi −1 ,
1.4
for almost every x ∈ Ω, for every z ∈ Rn , and for any i 1, 2, . . . , n, and there exists ν ∈ 0, ∞
such that
ν
n
|zi − zi |pi ≤
i1
n
ai x, z − ai x, zzi − zi ,
1.5
i1
for almost every x ∈ Ω, for any z, 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 ∈ W 1,pi Ω, and we
introduce
p 1,p Kψ,θi Ω v ∈ W 1,pi Ω : v ≥ ψ, a.e. and v − θ ∈ W0 i Ω .
1.6
Note that
1,pi W0
Ω v ∈ W01,1 Ω : Di v ∈ Lpi Ω for every i 1, 2, . . . , n .
1.7
The function ψ is an obstacle and θ determines the boundary values.
p p Definition 1.1. A solution to the Kψ,θi -obstacle problem is a function u ∈ Kψ,θi Ω such that
n
ai x, DuxDi vx − Di uxdx ≥ 0,
Ω i1
1.8
p whenever v ∈ Kψ,θi Ω.
Higher integrability property is important among the regularity theories of nonlinear
elliptic PDEs and systems, see the monograph 1 by Bensoussan and Frehse. Meyers and
Elcrat 2 first considered the higher integrability for weak solutions of 1.3 in 1975. Iwaniec
and Sbordone 3 obtained a regularity result for very weak solutions of the A-harmonic
equation 1.3 by using the celebrated Gehring’s Lemma. Global integrability for anisotropic
equation is contained in 4. As far as higher integrability of ∇u is concerned, in problems
with nonstandard growth a delicate interplay between the regularity with respect to x and
the growth with respect to ξ appears: see 5. For a global boundedness result of anisotropic
variational problems, see 6. For other related works, see 7. We refer the readers to the
classical books by Ladyženskaya and Ural’ceva 8, Morrey 9, Gilbarg and Trudinger 10
and Giaquinta 11 for some details of isotropic cases.
In the present paper, we consider integrability for solutions of anisotropic obstacle
problems of the A-harmonic equation 1.3, which show higher integrability of the boundary
International Journal of Mathematics and Mathematical Sciences
3
datum, and the obstacle force solutions u, have higher integrability as well. The idea of this
paper comes from 4, and the result can be considered as a generalization of 4, Theorem 2.1.
p p Theorem 1.2. Let u ∈ Kψ,θi Ω be a solution to the Kψ,θi obstacle problem and θ ∈ W 1,qi Ω,
qi ∈ pi , ∞, i 1, 2, . . . , n, 0 ≤ h ∈ Lqm Ω with qm maxi1,...,n qi , ψ ∈ −∞, ∞ is such that
1,q θ∗ max{ψ, θ} ∈ θ W0 i Ω. Moreover, p < n. Then
u ∈ θ∗ Ltweak Ω,
1.9
where
t
p∗
> p∗ ,
1 − bp∗ /p pm /pm − 1
1.10
and b is any number verifying
0 < b ≤ min
j1,...,n
1−
pj
qj
1−
1
pj
,
1.11
pm − 1 p
b<
.
pm p ∗
Remark 1.3. Take the obstacle function ψ to be minus infinity in Theorem 1.2, and the
condition 1.4 replaced by
|ai x, z| ≤ c2 1 |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 in 4.
2. Proof of the Main Theorem
p p Proof of Theorem 1.2. Let u ∈ Kψ,θi Ω be a solution to the Kψ,θi -obstacle problem. Take θ∗ 1,qi max{ψ, θ} ∈ θ W0
Ω. Let us consider L ∈ 0, ∞ and
⎧
⎪
⎪
⎨θ∗ − L, for u − θ∗ < −L,
v u,
for − L ≤ u − θ∗ ≤ L,
⎪
⎪
⎩θ L, for u − θ > L.
∗
∗
p 2.1
Then v ∈ Kψ,θi Ω. Indeed, for the second and the third cases of the above definition for v,
we obviously have v ≥ ψ, and for the first case, u − θ∗ < −L, we have θ∗ > u L ≥ ψ L; this
4
International Journal of Mathematics and Mathematical Sciences
implies v θ∗ − L ≥ ψ. Since u θ∗ θ on ∂Ω, then v u on ∂Ω, this implies v θ on ∂Ω. By
Definition 1.1, one has
0≤
n
{|u−θ∗ |>L} i1
n
{|u−θ∗ |>L} i1
ai x, DuxDi vx − Di uxdx
2.2
ai x, DuxDi θ∗ x − Di uxdx.
Monotonicity 1.5 allows us to write
ν
n i1
{|u−θ∗ |>L}
|Di ux − Di θ∗ x|pi dx
≤
2.3
n
ai x, Dux − ai x, Dθ∗ xDi ux − Di θ∗ xdx,
{|u−θ∗ |>L} i1
which together with 2.2 implies
ν
n {|u−θ∗ |>L}
i1
|Di ux − Di θ∗ x|pi dx
≤−
2.4
n
ai x, Dθ∗ Di ux − Di θ∗ xdx.
{|u−θ∗ |>L} i1
We now use anisotropic growth 1.4 and the Hölder inequality in 2.4, obtaining that
ν
n i1
{|u−θ∗ |>L}
≤−
|Di u − Di θ∗ |pi dx
n i1
≤ c2
ai x, Dθ∗ Di u − Di θ∗ dx
n i1
≤ c2
{|u−θ∗ |>L}
2.5
pi −1
{|u−θ∗ |>L}
h |Di θ∗ |
n
i1
|Di u − Di θ∗ |dx
pi −1/pi
{|u−θ∗ |>L}
h |Di θ∗ |pi dx
1/pi
{|u−θ∗ |>L}
|Di u − Di θ∗ |pi dx
.
Let ti be such that
pi < ti ≤ qi ,
2.6
International Journal of Mathematics and Mathematical Sciences
5
for every i 1, . . . , n; ti will be chosen later. We use the Hölder inequality as follows:
pi −1/pi
pi
{|u−θ∗ |>L}
h |Di θ∗ | dx
≤
2.7
pi −1/ti
|{|u − θ∗ | > L}|ti −pi pi −1/ti pi .
h |Di θ∗ |ti dx
{|u−θ∗ |>L}
The following proof is similar to that of 4, Theorem 2.1; we only list the necessary changes:
instead of 4, 3.14 by
pi −1/pi
{|u−θ∗ |>L}
h |Di θ∗ |pi dx
≤
pi −1/ti
ti
{|u−θ∗ |>L}
|{|u − θ∗ | > L}|b
h |Di θ∗ | dx
2.8
≤ M|{|u − θ∗ | > L}|b ,
where
M max
j1,...,n
Ω
h |Dj θ∗ |
tj
pj −1/tj
dx
2.9
< ∞,
and instead of 4, 3.19 we use anisotropic Sobolev Embedding Theorem for v − u,
p∗
Ω
|v − u| dx
≤ c∗
1/p∗
n i1
Ω
⎡
n
⎣
≤ c∗
i1
|Di v − u| dx
pi
1/pi 1/n
2.10
1/pi
{|u−θ∗ |>L}
|Di u − Di θ∗ |pi dx
⎤1/n
⎦
.
By |v − u| |u − θ∗ | − L1{|u−θ∗ |>L} , we obtain
{|u−θ∗ |>L}
p∗
|u − θ∗ | − L dx
1/p∗
Ω
∗
|v − u|p dx
1/p∗
.
2.11
Following the idea of the proof of Theorem 2.1 in 4, we complete the proof of Theorem 1.2.
6
International Journal of Mathematics and Mathematical Sciences
Acknowledgment
The authors would like to thank the referee for valuable suggestions. Supported by NSFC
10971224 and NSF of Hebei Province A2011201011.
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 für 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 with p, q growth,” 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. Ladyženskaya 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
Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1983.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Download