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, 2Qiao Jinjing,
3and Chu Yuming
41College 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:
wherep >1 and 0< α≤β <∞. The prototype of1.1is thep-harmonic equation:
div|∇u|p−2∇u0. 1.2
Suppose thatψis an arbitrary function inΩwith values in R∪ {±∞}, andθ∈W1,rΩwith
max{1, p−1}< r ≤p. Let
Kr ψ,θΩ
v∈W1,rΩ:v≥ψ a.e., andv−θ∈W01,rΩ. 1.3
The functionψis an obstacle andθdetermines the boundary values.
For anyu, v∈ Krψ,θΩ, we introduce the Hodge decomposition for|∇v−u|r−p∇v− u∈Lr/r−p1Ω, see1:
|∇v−u|r−p∇v−u ∇φ
v,uhv,u, 1.4
whereφv,u ∈ W01,r/r−p1Ωand hv,u ∈ Lr/r−p1Ω,Rnare a divergence-free vector field,
and the following estimates hold:
∇φv,u
r/r−p1≤c1∇v−urr−p1, 1.5
hv,ur/r−p1≤c1
p−r∇v−urr−p1, 1.6
wherec1c1n, pis some constant depending only onnandp.
Definition 1.1see2. A very weak solution to theKr
ψ,θ-obstacle problem is a functionu∈
Kr
ψ,θΩsuch that
Ω
Ax,∇u,|∇v−u|r−p∇v−udx≥
ΩAx,∇u, hv,udx, 1.7
wheneverv∈ Krψ,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 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 existsr1withmax{1, p−1}< r1< p, such that any very weak solutionuto theKr
ψ,θ-obstacle problem belongs toLs
∗
locΩ,s∗1/1/s−1/n, provided that0≤ψ ∈W 1,s
locΩ, 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 inWloc1,rΩis trivially inLt
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 existsr1 with max{1, p−1} < r1 < p, such that any very weak solution uto the Krψ,θ-obstacle problem belongs toLsloc∗Ω, provided that ψ ∈ Wloc1,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 Kp
ψ,θ-obstacle problem, then we have the following local regularity result.
Corollary 1.5. Suppose thatψ ∈ Wloc1,sΩ,1 < p < s < n. Then a solutionuto theKpψ,θ-obstacle problem belongs toLs∗
locΩ.
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 Kr
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 ∈ Wloc1,∞Ω, then a very weak solutionuto the Kr
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 ∈ Wloc1,∞Ω, a very weak solution u to theKr
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 weaksolutions 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},Ak {x∈Ω:
ux > k},Ak,t Ak∩Bt,Ak,t Ak∩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∈Wloc1,rΩ,ϕ0∈LqlocΩ, where1< r < nandqsatisfies
1< q < n
r. 1.10
Assume that the following integral estimate holds:
Ak,t
|∇u|rdx≤c
0
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∈Lqr
∗
loc Ω.
Lemma 1.9see14. Letfτbe a nonnegative bounded function defined for0 ≤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
fρ≤c2
AR−ρ−αB. 1.13
Definition 1.10see15. A functionux∈Wloc1,mΩbelongs to the classBΩ, γ, m, k0,if for
allk > k0,k0>0 and allBρBρx0,Bρ−ρσBρ−ρσx0,BR BRx0,one has
Ak,ρ−ρσ
|∇u|m
dx≤γ
σ−mρ−m
Ak,ρ
u−km
dxAk,ρ
, 1.14
forR/2≤ρ−ρσ < ρ < R,m < n, where|Ak,ρ|is then-dimensional Lebesgue measure of the setAk,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
max
BR/2
ux≤c, 1.15
in which the constantcis determined only by the quantitiesγ, m, k0, R,∇um1.
2. Local Regularity
Proof ofTheorem 1.4. Let u be a very weak solution to the Kr
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−φr
u−ψk
, 2.2
whereTkuis the usual truncation ofuat levelk≥0 defined in1.9andψkmax{ψ, Tku}.
Nowv∈ Krψ−Tku,θ−TkuΩ; indeed, sinceu∈ Krψ,θΩandφ∈C∞0Ω, then
v−θ−Tku u−θ−φr
u−ψk
∈W01,rΩ, v−ψ−Tku
u−ψ−φru−ψk
a.e. inΩ. Let
Ev, u φr∇ur−pφr∇u|∇v−uTku|r−p∇v−uTku, 2.4
By an elementary inequality16, Page 271,4.1,
|X|−εX− |Y|−εY≤2ε1ε
1−ε|X−Y|
1−ε, X, Y∈Rn, 0≤ε <1,
∇v∇u−Tku−φr∇
u−ψk
−rφr−1∇φu−ψk
,
2.5
one can derive that
|Ev, u| ≤2p−rp−r1 r−p1
φr∇ψk−rφr−1∇φ
u−ψk r−p1
. 2.6
We get from the definition ofEv, uthat
Ak,t
Ax,∇u,φr∇ur−pφr∇udx
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−udx.
2.7
Now we estimate the left-hand side of2.7. By conditionawe have
Ak,t
Ax,∇u,φr∇ur−pφr∇ud≥
Ak,τ
Ax,∇u,|∇u|r−p∇udx≥α Ak,τ
|∇u|rdx.
2.8
Sinceu−Tku, v∈ Krψ−Tku,θ−TkuΩ, then using the Hodge decomposition1.4, we get
|∇v−uTku|r−p∇v−uTku ∇φh, 2.9
and by1.6we have
hr/r−p1≤c1
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|r
dx≤
Ak,t
Ax,∇u, Ev, udx−
Ak,t
Ax,∇u, hdx
≤β2
p−rp−r1
r−p1 Ak,t|∇u|
p−1φr∇ψ
k−rφr−1∇φ
u−ψk r−p1
dx
β
Ak,t
|∇u|p−1|h|dx
≤β2
p−rp−r1
r−p1 Ak,t
|∇u|p−1φr∇ψ
kr−p1dx
β2
p−rp−r1
r−p1 Ak,t|∇u|
p−1
rφr−1∇φu−ψk r−p1
dx
β
Ak,t
|∇u|p−1|h|dx
≤β2
p−rp−r1
r−p1 Ak,t|∇u|
rdx
p−1/r
Ak,t ∇ψkr
dx
r−p1/r
β2
p−rp−r1
r−p1 Ak,t|∇u|
r
dx
p−1/r
×
Ak,t
rφr−1∇φu−ψk r
dx
r−p1/r
β
Ak,t
|∇u|r
dx
p−1/r
Ak,t
|h|r/r−p1dx
r−p1/r
.
Denotec3c3p, r 2p−rp−r1/r−p1. It is obvious that ifris sufficiently close top,
thenc3p, r≤2. By2.10and Young’s inequality
ab≤εapc4
ε, pbp, 1 p
1
p 1, a, b≥0, ε≥0, p≥1, 2.14
we can derive that
α
Ak,τ
|∇u|rdx≤βc
3
p, rε
Ak,t
|∇u|rdxβc
3
p, rc4
ε, p
Ak,t ∇ψkr
dx
βc3
p, rε
Ak,t
|∇u|r
dxβc3
p, rc4
ε, p
Ak,t
rφr−1∇φu−ψk r
dx
βc1ε
p−r
Ak,t
|∇u|r
dxβc1c4
ε, pp−r
Ω|∇v−uTku|
r
dx
≤βε2c3
p, rc1
p−r
Ak,t
|∇u|r
dxβc3
p, rc4
ε, p
Ak,t ∇ψkr
dx
βc3
p, rc4
ε, p
Ak,t
rφr−1∇φu−ψk r
dx
βc1c4
ε, pp−r
Ω|∇v−uTku|
rdx.
2.15
By the equality
∇v∇u−Tku−φr∇
u−ψk
−rφr−1∇φu−ψk
, 2.16
andv−uTku 0 forx∈Ω\Ak,t, then we have
Ω|∇v−uTku|
r
dx
Ak,t
φr∇u−ψk
rφr−1∇φu−ψk r
dx
≤2r−1
Ak,t
|∇u|r
dx
Ak,t ∇ψkr
dx
Ak,t
rφr−1∇φu−ψk r
dx
.
Finally we obtain that
Ak,τ
|∇u|r
dx≤ βε2c3
p, rc1
p−r2r−1βc 1c4
ε, pp−r
α Ak,t
|∇u|r
dx
βc3
p, rc4
ε, p2r−1βc 1c4
ε, pp−r
α Ak,t
∇ψkr
dx
βc3
p, rc4
ε, p2r−1βc 1c4
ε, pp−r
α Ak,t
rφr−1∇φu−ψ
k r
dx
≤ βε2c3
p, rc1
p−r2p−1βc 1c4
ε, pp−r
α Ak,t
|∇u|rdx
βc3
p, rc4
ε, p2p−1βc 1c4
ε, pp−r
α Ak,t
∇ψr
dx
βc3
p, rc4
ε, p2p−1βc 1c4
ε, pp−r α
2pp
t−τr Ak,t
|u|r
dx.
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, rc1
p−r2p−1βc1c4
ε, pp−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 Ak,R
∇ψr
dx c6
αt−τr Ak,R
|u|rdx,
2.20
wherec5 βc3p, rc4ε, p 2p−1βc1c4ε, pp−rwith εandr fixed to satisfy2.19, and c62ppc5. ApplyingLemma 1.9in2.20we conclude that
Ak,ρ
|∇u|r
dx≤ c2c5 α Ak,R
∇ψr
dx c2c6
αR−ρr Ak,R
|u|r
dx, 2.21
wherec2is the constant given byLemma 1.9. Thususatisfies inequality1.11withϕ0|∇ψ|r
3. Local Boundedness
Proof ofTheorem 1.6. Letube a very weak solution to theKr
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−φr
u−maxψ, tku
, 3.2
wheretku min{u, k}. Nowv∈ Krψ−tku,θ−tku; indeed, sinceu∈ Krψ,θΩandφ∈C∞0 Ω,
then
v−θ−tku u−θ−φr
u−maxψ, tku
∈W01,rΩ,
v−ψ−tku
u−ψ−φru−maxψ, tku
≥1−φru−ψ≥0 3.3
a.e. inΩ.
As in the proof ofTheorem 1.4, we obtain
Ak,τ
|∇u|r
dx≤ βε2c3
p, rc1
p−r2r−1βc 1c4
ε, pp−r
α Ak,t
|∇u|r
dx
βc3c4
ε, p2r−1βc 1c4
ε, pp−r
α Ak,t
∇max
ψ, tkurdx
βc3
p, rc4
ε, p2r−1βc 1c4
ε, pp−r α
×
Ak,t
rφr−1∇φu−max{ψ, tku} r
dx
≤ βε2c3
p, rc1
p−r2r−1βc 1c4
ε, pp−r
α Ak,t
|∇u|r
dx
βc3c4
ε, p2r−1βc 1c4
ε, pp−r
α Ak,t
∇ψr
dx
βc3
p, rc4
ε, p2r−1βc 1c4
ε, pp−r α
2pp
t−τr Ak,t
|u−k|r
dx.
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
Ak,τ
|∇u|r
dx≤θ
Ak,t
|∇u|r
dxc5 α Ak,R
∇ψr
dx αtc−6τr Ak,R
|u−k|r
ApplyingLemma 1.9, we conclude that
Ak,ρ
|∇u|rdx≤ c2c6
αR−ρr Ak,R
|u−k|rdxc2c5 α Ak,R
∇ψr
dx
≤ c2c6
αR−ρr Ak,R
|u−k|rdxc2c5c7 α
Ak,R,
3.6
wherec2is the constant given byLemma 1.9andc7∇ψpL∞Ω. Thusubelongs to the class
Bwithγmax{c2c6/α, c2c5c7/α}andmr.Lemma 1.11yields
max
BR/2
ux≤c. 3.7
This result together with the assumptionsu≥ψandψ ∈Wloc1,∞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.
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