Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 878769, 12 pages
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 Yuming4
1
College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
Hebei Provincial Center of Mathematics, Hebei Normal University, Shijiazhuang 050016, China
3
College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China
4
Faculty of Science, Huzhou Teachers College, Huzhou, Zhejiang 313000, China
2
Correspondence should be addressed to Gao Hongya, hongya-gao@sohu.com
Received 25 September 2009; Accepted 18 March 2010
Academic Editor: Yuming Xing
Copyright q 2010 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 in 1.5
and 1.6 are satisfied; see 1. A Lipschitz domain, for example, is a regular domain. We
consider the second-order divergence type elliptic equation also called A-harmonic equation
or Leray-Lions equation:
div Ax, ∇ux 0,
1.1
where Ax, ξ : Ω × Rn → Rn is a Carathéodory function satisfying the following conditions:
a Ax, ξ, ξ ≥ α|ξ|p ,
b |Ax, ξ| ≤ β|ξ|p−1 ,
c Ax, 0 0,
2
Journal of Inequalities and Applications
where p > 1 and 0 < α ≤ β < ∞. The prototype of 1.1 is the p-harmonic equation:
div |∇u|p−2 ∇u 0.
1.2
Suppose that ψ is an arbitrary function in Ω with values in R ∪ {±∞}, and θ ∈ W 1,r Ω with
max{1, p − 1} < r ≤ p. Let
Krψ,θ Ω v ∈ W 1,r Ω : v ≥ ψ a.e., and v − θ ∈ W01,r Ω .
1.3
The function ψ is an obstacle and θ determines the boundary values.
For any u, v ∈ Krψ,θ Ω, we introduce the Hodge decomposition for |∇v − u|r−p ∇v −
u ∈ Lr/r−p1 Ω, see 1:
|∇v − u|r−p ∇v − u ∇φv,u hv,u ,
1.4
1,r/r−p1
where φv,u ∈ W0
Ω and hv,u ∈ Lr/r−p1 Ω, Rn are a divergence-free vector field,
and the following estimates hold:
r−p1
∇φv,u ≤ c1 ∇v − ur
,
r/r−p1
1.5
r−p1
,
hv,u r/r−p1 ≤ c1 p − r ∇v − ur
1.6
where c1 c1 n, p is some constant depending only on n and p.
Definition 1.1 see 2. A very weak solution to the Krψ,θ -obstacle problem is a function u ∈
Krψ,θ Ω such that
Ω
Ax, ∇u, |∇v − u|r−p ∇v − u dx ≥
Ω
Ax, ∇u, hv,u dx,
1.7
whenever v ∈ Krψ,θ Ω.
Remark 1.2. If r p in Definition 1.1, then hv,u 0 by the uniqueness of the Hodge
decomposition 1.4, and 1.7 becomes
Ω
Ax, ∇u, ∇v − udx ≥ 0.
p
1.8
This is the classical definition for Kψ,θ -obstacle problem; see 3 for some details of solutions
p
of Kψ,θ -obstacle problem.
Journal of Inequalities and Applications
3
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 monograph 4
by Bensoussan and Frehse. Meyers and Elcrat 5 first considered the higher integrability for
weak solutions of 1.1 in 1975; see also 6. Iwaniec and Sbordone 1 obtained the regularity
result for very weak solutions of the A-harmonic 1.1 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 Martio 7 in 1994 by using the so-called reverse Hölder inequality.
Gao et al. 2 gave the definition for very weak solutions of obstacle problem of A-harmonic
1.1 and obtained the local and global higher integrability results. The local regularity results
for minima of functionals and solutions of elliptic equations have been obtained in 8. 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 function ψ ≥ 0. Li and Gao 13 generalized the result of 12 by obtaining the local
integrability result for very weak solutions of obstacle problem. The main result of 13 is the
following proposition.
Proposition 1.3. There exists r1 with max{1, p − 1} < r1 < p, such that any very weak solution u to
∗
1,s
Ω,
the Krψ,θ -obstacle problem belongs to Lsloc Ω, s∗ 1/1/s − 1/n, provided that 0 ≤ ψ ∈ Wloc
r < s < n, and r1 < r < min{p, n}.
Notice that in the above proposition we have restricted ourselves to the case r < n,
1,r
Ω is trivially in Ltloc Ω for every t > 1 by the
because when r ≥ n, every function in Wloc
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 1.3 is not necessary.
Theorem 1.4. There exists r1 with max{1, p − 1} < r1 < p, such that any very weak solution
∗
1,s
Ω, r < s < n, and
u to the Krψ,θ -obstacle problem belongs to Lsloc Ω, provided that ψ ∈ Wloc
r1 < r < min{p, n}.
As a corollary of the above theorem, if r p, that is, if we consider weak solutions of
p
Kψ,θ -obstacle problem, then we have the following local regularity result.
p
1,s
Corollary 1.5. Suppose that ψ ∈ Wloc
Ω, 1 < p < s < n. Then a solution u to the Kψ,θ -obstacle
∗
problem belongs to Lsloc Ω.
We omit the proof of this corollary. This corollary shows that the condition ψ ≥ 0 in the
main result of 12 is not necessary.
The second part of this paper considers local boundedness for very weak solutions
of Krψ,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 to 4. In this paper we consider very weak solutions
1,∞
Ω, then a very weak solution u to the
and show that if the obstacle function is ψ ∈ Wloc
r
Kψ,θ -obstacle problem is locally bounded.
4
Journal of Inequalities and Applications
Theorem 1.6. There exists r1 with max{1, p − 1} < r1 < p, such that for any r with r1 < r <
1,∞
Ω, a very weak solution u to the Krψ,θ -obstacle problem is locally
min{p, n} and any ψ ∈ Wloc
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. If x0 ∈ Ω and t > 0, then Bt denotes the ball of radius
t centered at x0 . For a function ux and k > 0, let Ak {x ∈ Ω : |ux| > k}, Ak {x ∈ Ω :
ux > k}, Ak,t Ak ∩ Bt , Ak,t Ak ∩ Bt . Moreover if s < n, s∗ is always the real number
satisfying 1/s∗ 1/s − 1/n. Let Tk u be the usual truncation of u at level k > 0, that is,
Tk u max{−k, min{k, u}}.
1.9
Let tk u min{u, k}.
We recall two lammas which will be used in the proof of Theorem 1.4.
q
1,r
Ω, ϕ0 ∈ Lloc Ω, where 1 < r < n and q satisfies
Lemma 1.8 see 8. Let u ∈ Wloc
1<q<
n
.
r
1.10
Assume that the following integral estimate holds:
−α
r
|∇u| dx ≤ c0
Ak,t
r
ϕ0 dx t − τ
Ak,t
|u| dx ,
1.11
Ak,t
for every k ∈ N and R0 ≤ τ < t ≤ R1 , where c0 is a real positive constant that depends only on
qr∗
N, q, r, R0 , R1 , |Ω| and α is a real positive constant. Then u ∈ Lloc Ω.
Lemma 1.9 see 14. Let fτ be a nonnegative bounded function defined for 0 ≤ R0 ≤ t ≤ R1 .
Suppose that for R0 ≤ τ < t ≤ R1 one has
fτ ≤ At − τ−α B θft,
1.12
where A, B, α, θ are nonnegative constants and θ < 1. Then there exists a constant c2 c2 α, θ,
depending only on α and θ, such that for every ρ, R, R0 ≤ ρ < R ≤ R1 one has
−α
B .
f ρ ≤ c2 A R − ρ
We need the following definition.
1.13
Journal of Inequalities and Applications
5
1,m
Ω belongs to the class BΩ, γ, m, k0 , if for
Definition 1.10 see 15. A function ux ∈ Wloc
all k > k0 , k0 > 0 and all Bρ Bρ x0 , Bρ−ρσ Bρ−ρσ x0 , BR BR x0 , one has
m
Ak,ρ−ρσ
|∇u| dx ≤ γ σ
−m −m
ρ
u − k dx Ak,ρ ,
m
Ak,ρ
1.14
for R/2 ≤ ρ − ρσ < ρ < R, m < n, where |Ak,ρ | is the n-dimensional Lebesgue measure of the
set Ak,ρ .
We recall a lemma from 15 which will be used in the proof of Theorem 1.6.
Lemma 1.11 see 15. Suppose that ux is an arbitrary function belonging to the class
BΩ, γ, m, k0 and BR ⊂⊂ Ω. Then one has
max ux ≤ c,
BR/2
1.15
in which the constant c is determined only by the quantities γ, m, k0 , R, ∇um1 .
2. Local Regularity
Proof of Theorem 1.4. Let u be a very weak solution to the Krψ,θ -obstacle problem. By
Lemma 1.8, it is sufficient to prove that u satisfies the inequality 1.11 with α r. Let
BR1 ⊂⊂ Ω and 0 < R0 ≤ τ < t ≤ R1 be arbitrarily fixed. Fix a cut-off function φ ∈ C0∞ BR1 such
that
supp φ ⊂ Bt ,
0 ≤ φ ≤ 1, φ 1 in Bτ , ∇φ ≤ 2t − τ−1 .
2.1
Consider the function
v u − Tk u − φr u − ψk ,
2.2
where Tk u is the usual truncation of u at level k ≥ 0 defined in 1.9 and ψk max{ψ, Tk u}.
Now v ∈ Krψ−Tk u,θ−Tk u Ω; indeed, since u ∈ Krψ,θ Ω and φ ∈ C0∞ Ω, then
v − θ − Tk u u − θ − φr u − ψk ∈ W01,r Ω,
v − ψ − Tk u u − ψ − φr u − ψk ≥ 1 − φr u − ψ ≥ 0,
2.3
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Journal of Inequalities and Applications
a.e. in Ω. Let
r−p
Ev, u φr ∇u φr ∇u |∇v − u Tk u|r−p ∇v − u Tk u,
2.4
By an elementary inequality 16, Page 271, 4.1,
−ε
|X| X − |Y |−ε Y ≤ 2ε 1 ε |X − Y |1−ε , X, Y ∈ Rn , 0 ≤ ε < 1,
1−ε
∇v ∇u − Tk u − φr ∇ u − ψk − rφr−1 ∇φ u − ψk ,
2.5
one can derive that
|Ev, u| ≤ 2p−r
r−p1
p − r 1 r
.
φ ∇ψk − rφr−1 ∇φ u − ψk r−p1
2.6
We get from the definition of Ev, u that
Ak,t
r−p
Ax, ∇u, φr ∇u φr ∇u dx
Ax, ∇u, Ev, udx
Ak,t
−
Ax, ∇u, |∇v − u Tk u|r−p ∇v − u Tk u dx
Ak,t
2.7
Ax, ∇u, Ev, udx
Ak,t
−
Ax, ∇u, |∇v − u|r−p ∇v − u dx.
Ak,t
Now we estimate the left-hand side of 2.7. By condition a we have
r−p
Ax, ∇u, φr ∇u φr ∇ud ≥
Ak,t
Ax, ∇u, |∇u|
r−p
|∇u|r dx.
∇udx ≥ α
Ak,τ
Ak,τ
2.8
Since u − Tk u, v ∈ Krψ−Tk u,θ−Tk u Ω, then using the Hodge decomposition 1.4, we get
|∇v − u Tk u|r−p ∇v − u Tk u ∇φ h,
2.9
r−p1
.
hr/r−p1 ≤ c1 p − r ∇v − u Tk ur
2.10
and by 1.6 we have
Journal of Inequalities and Applications
7
Thus we derive, by Definition 1.1, that
Ax, ∇u − Tk u, |∇v − u Tk u|r−p ∇v − u Tk u dx
Ω
≥ Ax, ∇u − Tk u, hdx.
2.11
Ω
This means, by condition c, that
Ax, ∇u, |∇v − u|r−p ∇v − udx ≥
Ak,t
Ax, ∇u, hdx.
2.12
Ak,t
Combining the inequalities 2.7, 2.8, and 2.12, and using Hölder’s inequality and
condition b, we obtain
r
|∇u| dx ≤
α
Ak,τ
Ax, ∇u, Ev, udx −
Ak,t
p−r
Ax, ∇u, hdx
Ak,t
p−r1
r−p1
dx
≤β
|∇u|p−1 φr ∇ψk − rφr−1 ∇φ u − ψk r−p1
Ak,t
β
|∇u|p−1 |h|dx
2
Ak,t
r−p1
2p−r p − r 1
≤β
dx
|∇u|p−1 φr ∇ψk r−p1
Ak,t
2p−r p − r 1
r−p1
β
dx
|∇u|p−1 rφr−1 ∇φ u − ψk r−p1
Ak,t
|∇u|p−1 |h|dx
β
2.13
Ak,t
p−1/r r−p1/r
r
2p−r p − r 1
r
∇ψk dx
≤β
|∇u| dx
r−p1
Ak,t
Ak,t
p−1/r
2p−r p − r 1
r
β
|∇u| dx
r−p1
Ak,t
r
r−1 rφ ∇φ u − ψk dx
×
r−p1/r
Ak,t
p−1/r r−p1/r
r/r−p1
r
β
|∇u| dx
Ak,t
|h|
Ak,t
dx
.
8
Journal of Inequalities and Applications
Denote c3 c3 p, r 2p−r p − r 1/r − p 1. It is obvious that if r is sufficiently close to p,
then c3 p, r ≤ 2. By 2.10 and Young’s inequality
ab ≤ εap c4 ε, p bp ,
1 1
1,
p p
a, b ≥ 0, ε ≥ 0, p ≥ 1,
2.14
we can derive that
r
|∇u| dx βc3 p, r c4 ε, p
|∇u| dx ≤ βc3 p, r ε
α
Ak,τ
r
Ak,t
βc3 p, r ε
∇ψk r dx
Ak,t
|∇u|r dx βc3 p, r c4 ε, p
Ak,t
βc1 ε p − r
r
r−1 rφ ∇φ u − ψk dx
Ak,t
|∇u|r dx βc1 c4 ε, p p − r
Ak,t
≤ βε 2c3 p, r c1 p − r
Ω
|∇v − u Tk u|r dx
|∇u| dx βc3 p, r c4 ε, p
r
Ak,t
βc3 p, r c4 ε, p
∇ψk r dx
Ak,t
r
r−1 rφ ∇φ u − ψk dx
Ak,t
βc1 c4 ε, p p − r
Ω
|∇v − u Tk u|r dx.
2.15
By the equality
∇v ∇u − Tk u − φr ∇ u − ψk − rφr−1 ∇φ u − ψk ,
2.16
and v − u Tk u 0 for x ∈ Ω \ Ak,t , then we have
Ω
r
r φ ∇ u − ψk rφr−1 ∇φ u − ψk dx
|∇v − u Tk u|r dx Ak,t
∇ψk r dx r
|∇u| dx ≤2
r−1
Ak,t
Ak,t
r
r−1 rφ ∇φ u − ψk dx .
Ak,t
2.17
Journal of Inequalities and Applications
9
Finally we obtain that
βε 2c3 p, r c1 p − r 2r−1 βc1 c4 ε, p p − r
|∇u| dx ≤
|∇u|r dx
α
Ak,t
βc3 p, r c4 ε, p 2r−1 βc1 c4 ε, p p − r
∇ψk r dx
α
Ak,t
r−1
βc3 p, r c4 ε, p 2 βc1 c4 ε, p p − r
r
r−1 rφ ∇φ u − ψk dx
α
Ak,t
βε 2c3 p, r c1 p − r 2p−1 βc1 c4 ε, p p − r
≤
|∇u|r dx
α
Ak,t
r
βc3 p, r c4 ε, p 2p−1 βc1 c4 ε, p p − r
∇ψ dx
α
Ak,t
βc3 p, r c4 ε, p 2p−1 βc1 c4 ε, p p − r
2p p
|u|r dx.
α
t − τr Ak,t
2.18
r
Ak,τ
The last inequality holds since |u − ψk | ≤ |u| a.e. in Ak,t . Now we want to eliminate the first
term in the right-hand side containing ∇u. Choose ε small enough and r sufficiently close to
p such that
βε 2c3 p, r c1 p − r 2p−1 βc1 c4 ε, p p − r
< 1,
θ
α
2.19
and let ρ, R be arbitrarily fixed with R0 ≤ ρ < R ≤ R1 . Thus, from 2.18, we deduce that for
every τ and t such that ρ ≤ τ < t ≤ R, we have
|∇u|r dx ≤ θ
Ak,τ
|∇u|r dx Ak,t
c5
α
Ak,R
r
∇ψ dx c6
αt − τr
|u|r dx,
Ak,R
2.20
where c5 βc3 p, rc4 ε, p 2p−1 βc1 c4 ε, pp − r with ε and r fixed to satisfy 2.19, and
c6 2p pc5 . Applying Lemma 1.9 in 2.20 we conclude that
c2 c5
|∇u| dx ≤
α
Ak,ρ
r
r
∇ψ dx c2 c6 r
α R−ρ
Ak,R
|u|r dx,
Ak,R
2.21
where c2 is the constant given by Lemma 1.9. Thus u satisfies inequality 1.11 with ϕ0 |∇ψ|r
and α r. Theorem 1.4 follows from Lemma 1.8.
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Journal of Inequalities and Applications
3. Local Boundedness
Proof of Theorem 1.6. Let u be a very weak solution to the Krψ,θ -obstacle problem. Let BR1 ⊂⊂ Ω
and R1 /2 ≤ τ < t ≤ R1 be arbitrarily fixed. Fix a cut-off function φ ∈ C0∞ BR1 such that
0 ≤ φ ≤ 1, φ 1 in Bτ , ∇φ ≤ 2t − τ−1 .
supp φ ⊂ Bt ,
3.1
Consider the function
v u − tk u − φr u − max ψ, tk u ,
3.2
where tk u min{u, k}. Now v ∈ Krψ−tk u,θ−tk u ; indeed, since u ∈ Krψ,θ Ω and φ ∈ C0∞ Ω,
then
v − θ − tk u u − θ − φr u − max ψ, tk u ∈ W01,r Ω,
v − ψ − tk u u − ψ − φr u − max ψ, tk u ≥ 1 − φr u − ψ ≥ 0
3.3
a.e. in Ω.
As in the proof of Theorem 1.4, we obtain
βε 2c3 p, r c1 p − r 2r−1 βc1 c4 ε, p p − r
|∇u| dx ≤
|∇u|r dx
α
Ak,t
r−1
βc3 c4 ε, p 2 βc1 c4 ε, p p − r
∇ max ψ, tk u r dx
α
Ak,t
r
Ak,τ
βc3 p, r c4 ε, p 2r−1 βc1 c4 ε, p p − r
α
r
r−1
×
rφ ∇φu − max{ψ, tk u} dx
3.4
Ak,t
≤
βε 2c3 p, r c1 p − r 2r−1 βc1 c4 ε, p p − r
|∇u|r dx
α
Ak,t
r−1
r
βc3 c4 ε, p 2 βc1 c4 ε, p p − r
∇ψ dx
α
Ak,t
βc3 p, r c4 ε, p 2r−1 βc1 c4 ε, p p − r
2p p
|u − k|r dx.
α
t − τr Ak,t
Choose ε small enough and r1 sufficiently close to p such that 2.19 holds. Let ρ, R be
arbitrarily fixed with R1 /2 ≤ ρ < R ≤ R1 . Thus from 3.4 we deduce that for every τ and
t such that R1 /2 ≤ τ < t ≤ R1 , we have
c5
|∇u| dx ≤ θ
|∇u| dx α
Ak,t
r
Ak,τ
r
Ak,R
r
∇ψ dx c6
αt − τr
Ak,R
|u − k|r dx.
3.5
Journal of Inequalities and Applications
11
Applying Lemma 1.9, we conclude that
r
Ak,ρ
c2 c6
|∇u| dx ≤ r
α R−ρ
c2 c 6
≤ r
α R−ρ
c2 c5
|u − k| dx α
Ak,R
r
Ak,R
r
∇ψ dx
c2 c5 c7 |u − k|r dx Ak,R ,
α
Ak,R
3.6
p
where c2 is the constant given by Lemma 1.9 and c7 ∇ψL∞ Ω . Thus u belongs to the class
B with γ max{c2 c6 /α, c2 c5 c7 /α} and m r. Lemma 1.11 yields
max ux ≤ c.
BR/2
3.7
1,∞
This result together with the assumptions u ≥ ψ and ψ ∈ Wloc
Ω yields 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 NSFC 10971224 and NSF of Hebei
Province 07M003. The third author is supported by NSF of Zhejiang province Y607128
and NSFC 10771195.
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