Global Hoelder continuity of solutions to quasilinear equations with Morrey data

0.39 MB·English

by  Sun-Sig Byun

GLOBAL HO¨LDER CONTINUITY OF SOLUTIONS TO
QUASILINEAR EQUATIONS WITH MORREY DATA
5
1
SUN-SIGBYUN,DIANK.PALAGACHEV,ANDPILSOOSHIN
0
2
Abstract. Wedeal withgeneral quasilineardivergence-formcoerciveopera-
n
tors whose prototype is the m-Laplacean operator. The nonlinear terms are
a
givenbyCarath´eodoryfunctions andsatisfycontrolledgrowthstructurecon-
J
ditionswithdatabelongingtosuitableMorreyspaces. Thefairlynon-regular
5
boundary of the underlying domain is supposed to satisfy a capacity density
2
conditionwhichallowsdomainswithexteriorcorkscrewproperty.

We prove global boundedness and H¨older continuity up to the boundary
P] fortheweaksolutionsofsuchequations,generalizingthiswaytheclassicalLp-
resultofLadyzhenskaya andUral’tsevatothesettings oftheMorreyspaces.
A
.
h
t
a
m
1. Introduction
[
The general aim of the present article is to get sufficient conditions ensuring

1 boundedness and Ho¨lder continuity up to the boundary for the weak solutions to
v
general quasilinear equations with discontinuous ingredients which are controlled
2
within the Morrey functional scales. Precisely, we deal with weak solutions u ∈
9
1 W1,m(Ω) of the Dirichlet problem
0
6
0 div a(x,u,Du) =b(x,u,Du) in Ω
(1.1)
.
1 (u=(cid:0)0 (cid:1) on ∂Ω,
0
5 where Ω ⊂ Rn, n ≥ 2, is a bounded domain with generally non-smooth boundary,
1 m∈(1,n],anda: Ω×R×Rn →Rnandb: Ω×R×Rn →RareCarath´eodorymaps.
v: Letusstressthereaderattentionattheverybeginningthatprototypesofthequasi-
i linear equations studied are these for the m-Laplace operator div |Du|m−2Du
X
m−2
r with m>1, or these for the m-area type operator div A+|Du|2 (cid:0)2 Du with(cid:1)
a
m≥2 and A>0. (cid:16)(cid:0) (cid:1) (cid:17)
Regarding the nonlinear terms in (1.1), we assume controlled growths with re-
spect to u and Du, that is,
|a(x,u,Du)|=O ϕ(x)+|u|ℓ(mm−1) +|Du|m−1 ,
|b(x,u,Du)|=O (cid:16)ψ(x)+|u|ℓ−1+|Du|m(ℓℓ−1) (cid:17)
(cid:16) (cid:17)

Date:January27,2015.
2010 Mathematics Subject Classification. Primary 35J60, 35B65; Secondary 35R05, 35B45,
35J92,46E30.
Key words and phrases. Quasilinear elliptic operator; Coercive boundary value problem; m-
Laplacean; Weak solution; Controlled growths; Natural growths; Morrey space; Variational ca-
pacity;Essentialboundedness; H¨oldercontinuity.
1
2 S.-S.BYUN,D.K.PALAGACHEV,P.SHIN
as |z|,|Du| → ∞, where ℓ is the Sobolev conjugate of m, and coercivity of the
differential operator considered
a(x,u,Du)·Du≥γ|Du|m−Λ|u|ℓ−Λϕ(x)mm−1
withnon-negativefunctionsϕandψandconstantsγ >0andΛ≥0.Itisworthnot-
m nm
ingthatϕ∈Lm−1(Ω)andψ ∈Lnm+m−n(Ω), togetherwiththecontrolledgrowths,
aretheminimal hypothesesonthedataunderwhichtheconceptofW1,m(Ω)-weak
0
solution to (1.1) makes sense. In what follows, we will assume that ϕ and ψ are
non-negative measurable functions belonging to suitable Morrey spaces. Namely,
we suppose
ϕ∈Lp,λ(Ω) with p> m , λ∈(0,n) and (m−1)p+λ>n
(1.2) m−1
ψ ∈Lq,µ(Ω) with q > mn , µ∈(0,n) and mq+µ>n.
mn+m n
−
The non-regular boundary of Ω will be assumed to satisfy a density condition
expressedin terms of variational P-capacity for some P ∈(1,m) (see (2.1) below),
whichrequiresthecomplementRn\ΩtobeuniformlyP-thick. Thisnotionisanat-
uralgeneralizationofthemeasuredensitycondition,knownalsoas(A)-condition of
LadyzhenskayaandUraltseva(cf. [12,13,14]), whichholds forinstance wheneach
point of ∂Ω supports the exterior cone property, excluding this way exterior spikes
on ∂Ω. In that sense, the uniform P-thickness condition is satisfied by domains
with C1-smooth or Lipschitz continuous boundaries, but it holds also when ∂Ω is
flatinthe senseofReifenberg,including this wayboundarieswithfractalstructure
suchas the vonKochsnowflake. Anyway,the classofdomainsverifyingthe capac-
ity density condition (2.1) goes beyond these common examples and contains for
example sets with boundaries which support the uniform corkscrew condition.
The regularity problem for solutions to (1.1) has been a long-standing problem
in the PDEs theory, related to the Hilbert 19th Problem. In particular, the task
to get Ho¨lder continuity of the weak solutions under very general hypotheses on
thedataisafirststeptowardsdevelopingrelevantsolvabilityandregularitytheory
for (1.1) in the framework of various functional scales (see for instance [3, 4, 21]
and the references therein). In case when (1.1) is the Euler–Lagrange equation of
a given functional F that is the problem of regularity of the minimizers of F and
this links (1.1) to important equations from differential geometry or mathematical
physics, such as Gunzburg–Landau, nonlinear Schr¨odinger, non-Newtonian fluids
and so on.
TheHilbert19thProblemhasbeenbrilliantlysolvedbyDeGiorgiin[5]forW1,2-
0
weak solutions to linear differential operators over Lipschitz continuous domains
when m = 2, ϕ ∈ Lp with p > n and ψ ∈ Lq with 2q > n, and this provided the
initialbreakthroughinthemoderntheoryofquasilinearequationsinmorethantwo
independent variables. The De Giorgi result was extended to linear equations in
thenon-Lpsettings(i.e.,whenasortof (1.2)holds)byMorreyin[18]andLewyand
Stampacchia in [16] to equations with measures at the right-hand side, assuming
ϕ∈L2,λ, ψ ∈L1,µ with λ,µ>n−2. Moving to the quasilinear equation (1.1), we
disposeoftheseminalLp-resultofSerrin[25],whichprovidesinterior boundedness
andHo¨ldercontinuityoftheW1,m-weaksolutionsto(1.1)inthesub-controlled case
0
whenthe nonlinearities growas |u|m 1+|Du|m 1, andthe behaviour with respect
− −
to x of a(x,u,Du) and b(x,u,Du) is controlled in terms of ϕ and ψ, respectively,
QUASILINEAR EQUATIONS WITH MORREY DATA 3
with
ϕ∈Lp(Ω) with p> m , (m−1)p>n
(1.3) m−1
ψ ∈Lq(Ω) with q > mn , mq >n.
mn+m n
−
Global boundedness of the W1,m-weak solutions to (1.1) with general nonlin-
0
earities of controlled growths has been obtained by Ladyzhenskaya and Ural’tseva
in [12] under the hypotheses (1.3) and for domains satisfying the measure den-
sity (A)-condition. Assuming natural growths of the data that is, a(x,u,Du) =
O(ϕ(x)+|Du|m 1)andb(x,u,Du)=O(ψ(x)+|Du|m) and(1.3), Ladyzhenskaya
− (cid:0)
and Ural’tseva proved later in [13] Ho¨lder continuity up to the boundary for the
(cid:1)
bounded weak solutions of (1.1), and Gariepy and Ziemer extended in [6] their re-
sult to domains with P-thick complements. It was Trudinger [26] the first to get
global Ho¨lder continuity of the bounded solutions in the non-Lp settings under the
natural structurehypotheses ofLadyzhenskayaandUral’tsevawithϕ∈Ln/(m 1),ε,
−
ψ ∈ Ln/m,ε for a small ε > 0, while Lieberman derived in [17] a very general re-
sult on interior Ho¨lder continuity when ϕ and ψ are suitable measures. We refer
the author also to the works by Rakotoson [23], Rakotoson and Ziemer [24] and
Zamboni [27] for various interior regularity results regarding the problem (1.1).
Thispaperisanaturalcontinuationof[2]whereboundednesshasbeenprovedfor
(1.1)withMorreydatainthecasem=2underthetwo-sided(A)conditionon∂Ω.
Here we derive global boundedness (Theorem 2.1) and Ho¨lder continuity up to the
boundary (Theorem 2.3) for each W1,m(Ω)-weak solution of the coercive Dirichlet
0
problem(1.1)overdomainswithP-thick complementsassumingcontrolled growths
ofthenonlinearitiesandMorreydataϕandψsatisfying(1.2). Apartfromthemore
general class of domains considered, we extend this way the classical Lp-results of
LadyzhenskayaandUral’tseva[12,13,14]to the non-Lp-settings by weakeningthe
hypotheses on ϕ and ψ to the scales of Morrey type. A comparison between (1.2)
and(1.3)showsthatthedecreaseofthedegreespandq ofLebesgueintegrabilityof
thedataϕandψisattheexpenseofincreaseoftheMorreyexponentsλandµ,and
the rangeof these variations is alwayscontrolledby the relations (m−1)p+λ>n
and mq + µ > n. Indeed, in the particular case λ = µ = 0 and domains with
exteriorcone property,our results reduce to these of LadyzhenskayaandUraltseva
[12,13,14]. However,ourTheorems2.1and2.3generalizesubstantiallythe results
in [12, 13, 14] because even if (m−1)p ≤ n and mq ≤ n, there exist functions
ϕ ∈ Lp,λ with (m−1)p+λ > n and ψ ∈ Lq,µ with mq+µ > n for which (1.2)
hold, but ϕ ∈/ Lp′ ∀p > n/(m−1) and ψ ∈/ Lq′ ∀q > n/m and therefore (1.3)
′ ′
fail. Moreover, as will be seen in Section 4 below, the controlled growths and
the restrictions (1.2) on the Sobolev–Morrey exponents are optimal for the global
boundedness and the subsequent Ho¨lder continuity of the weak solutions to (1.1).
The paper is organized as follows. In Section 2 we start with introducing the
concept of P-thickness and discuss its relations to the measure density property
of ∂Ω. We list in a detailed way the hypotheses imposed on the data of (1.1) and
state the main results of the paper. Section 3 collects various auxiliary results
which form the analytic heart of our approach. Of particular interest here is the
Gehring–Giaquinta–ModicatypeLemma3.8thatassertsbetter integrability forthe
gradientofthe weaksolutionoverdomainswithP-thickcomplements,aparticular
case of which is due to Kilpel¨ainen and Koskela [11]. The proof of the global
boundedness result(Theorem2.1)isgiveninSection4. Ourtechniquereliesonthe
4 S.-S.BYUN,D.K.PALAGACHEV,P.SHIN
DeGiorgiapproachtotheboundednessasadaptedbyLadyzhenskayaandUraltseva
(cf. [14,ChapterIV])toquasilinearequations. Namely,usingthecontrolledgrowth
assumptions, we get exact decay estimates for the total mass of the weak solution
taken over its level sets. However, unlike the Lp-approach of Ladyzhenskaya and
Uraltseva, the mass we have to do with is taken with respect to a positive Radon
m
measure M, which depends not only on the Lebesgue measure, but also on ϕm−1,
ψ anda suitable power ofthe weaksolutionitself. Thanks to the hypotheses (1.2),
the measure M allows to employ very precise inequalities of trace type due to
D.R. Adams [1] and these lead to a bound of the M-mass of u in terms of the
m-energy of u. At this point we combine the controlled growth conditions with
the better integrability of the gradient in order to estimate the m-energy of u in
terms of small multiplier of the same quantity plus a suitable power of the level
set M-measure. The global boundedness of the weak solution then follows by a
classical result known as Hartman–Stampacchia maximum principle. At the end
of Section 4 we show sharpness of the controlled growths hypotheses as well as
of (1.2) on the level of explicit examples built on quasilinear operators with m-
Laplacean principal part. Section 5 is devoted to the proof of the global Ho¨lder
continuity asclaimedinTheorem2.3. Indeed,theboundednessoftheweaksolution
is guaranteed by Theorem 2.1 and the fine results obtained by Lieberman in [17]
apply to infer interior Ho¨lder continuity. To extend it up to the boundary of Ω,
weadoptto oursituationthe approachofGariepyandZiemerfrom[6]whichrelies
on the Moser iteration technique in obtaining growth estimates for the gradient
of the solution. The crucial step here is ensured by Lemma 5.1 which combines
with the P-thickness conditionin orderto getestimate for the oscillationof u over
smallballs centeredon∂Ω in terms ofa suitable positive power ofthe radius. Just
for the sake of simplicity, we proved Theorem 2.3 under the controlled growths
hypotheses. Following the same arguments, it is easy to see that the global Ho¨lder
continuity result still holds true for the bounded weak solutions of (1.1) if one
assumes the natural structure conditions of Ladyzhenskaya and Ural’tseva instead
of the controlled ones (cf. Theorem 5.2).
Acknowledgments. S.-S. Byun was supported by the National Research Foun-
dationof Korea(NRF) grantfunded by the Koreagovernment(MSIP) (No. 2009–
2012R1A2A2A01047030). D.K. Palagachev is member of the Gruppo Nazionale
per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the
Istituto Nazionale di Alta Matematica (INdAM).
2. Hypotheses and Main Results
Throughout the paper, we will use standard notations and will assume that the
functions and sets considered are measurable.
We denote by B (x) (or simply B if there is no ambiguity) the n-dimensional
ρ ρ
open ball with center x∈Rn and radius ρ. The Lebesgue measure of a measurable
set E ⊂ Rn will be denoted by |E| while, for any integrable function u defined on
a set A, its integral averageis given by
1
u :=− u(x)dx= u(x)dx.
A
|A|
ZA ZA
We will denote by C (Ω) the space of infinitely differentiable functions over a
0∞
bounded domain Ω ⊂ Rn with compact support contained in that domain, and
QUASILINEAR EQUATIONS WITH MORREY DATA 5
Lp(Ω)standsforthestandardLebesguespacewithagivenp∈[1,∞].TheSobolev
spaceW1,p(Ω)isdefined,asusual,bythecompletionofC (Ω)withrespecttothe
0 0∞
norm
kuk :=kuk +kDuk
W1,p(Ω) Lp(Ω) Lp(Ω)
for p∈[1,∞).
Given s∈[1,∞) and θ ∈[0,n], the Morrey space Ls,θ(Ω) is the collection of all
functions u∈Ls(Ω) such that
1/s
kuk := sup ρ θ |u(x)|s dx <∞.
Ls,θ(Ω) −
x0∈Ω, ρ>0 ZBρ(x0)∩Ω !
The space Ls,θ(Ω), equipped with the norm k·k is Banach space and the
Ls,θ(Ω)
limit cases θ =0 and θ =n give rise, respectively, to Ls(Ω) and L (Ω).
∞
Let Ω⊂Rn be a bounded domain with n≥2. In order to set down the require-
ments on ∂Ω, we need to recall the concept of variational p-capacity of a set for
1<p<∞. Thus, given a compact set C ⊂Ω, its p-capacity is defined as
Cap (C,Ω)=inf |Dg|p dx
p
g ZΩ
where the infimum is taken over all functions g ∈ C (Ω) such that g = 1 in C. If
0∞
U ⊂Ω is an open set, then
Cap (U,Ω)= sup Cap (C,Ω), C is compact,
p p
C U
⊂
while, if E ⊂Ω is an arbitrary set, then
Cap (E,Ω)= inf Cap (U,Ω), U is open.
p p
E U Ω
⊂ ⊂
In particular, if E ⊂E ⊂Ω ⊂Ω then
′ ′
Capp(E,Ω)≤Capp(E′,Ω′)
and, in case of two concentric balls B and B with R>r, the next formula
R r
Capp(Br,BR)=Crn−p
is known for p > 1, where C > 0 depends on n, p and R/r (see [9, Chapter 2] for
more details).
In the sequelwe willsuppose thatthe complementRn\ΩofΩ satisfies the next
uniform P-thickness condition for some P ∈ (1,m): there exist positive constants
A and r such that
Ω 0
(2.1) Cap Rn\Ω ∩B (x),B (x) ≥A Cap B (x),B (x)
P r 2r Ω P r 2r
for all x∈Rn\Ω(cid:0)(cid:0)and all(cid:1)r ∈(0,r0). (cid:1) (cid:0) (cid:1)
Let us point out that replacing the capacity above with the Lebesgue measure,
(2.1)reducestothemeasuredensitycondition(the(A)-condition ofLadyzhenskaya
andUral’tseva)whichholdsforinstancewhenΩsupportstheuniformexteriorcone
property. IfagivensetE satisfiesthemeasuredensityconditionthenitisuniformly
p-thick for each p > 1, whereas each nonempty set is uniformly p-thick if p > n.
Further on, a uniformly q-thick set is also uniformly p-thick for all p ≥ q and, as
proved in [15], the uniformly p-thick sets have a deep self-improving property to
be uniformly q-thick for some q < p, depending on n, p and the constant of the
p-thickness. In this sense, it is not restrictive to ask P < m in (2.1) since even
6 S.-S.BYUN,D.K.PALAGACHEV,P.SHIN
if Rn \Ω were m-thick, the existence of a P < m verifying (2.1) is ensured by
[15]. Yet another example of uniformly p-thick sets for all p > 1 is given by those
satisfying the uniform corkscrew condition: a set E is uniformly corkscrew if there
exist constants C >0 and r >0 such that for any x∈E and any r∈(0,r ) there
0 0
is a point y ∈B (x)\E with the property that B (y)⊂Rn\E.
r r/C
Turning back to the Dirichlet problem (1.1), the nonlinearities considered are
givenbytheCarath´eodorymapsa: Ω×R×Rn→Rnandb: Ω×R×Rn →R,where
a(x,z,ξ) = a1(x,z,ξ),··· ,an(x,z,ξ) . In other words, the functions ai(x,z,ξ)
and b(x,z,ξ) are measurable with respect to x∈Ω for all (z,ξ)∈R×Rn and are
continuousw(cid:0)ithrespectto z ∈R andξ(cid:1)∈Rn foralmostall(a.a.) x∈Ω. Moreover,
we suppose:
• Controlled growth conditions: There exist a constant Λ >0 and non-negative
functions ϕ ∈ Lp,λ(Ω) with p > m , λ ∈ (0,n) and (m − 1)p + λ > n, and
m 1
ψ ∈Lq,µ(Ω) with q > mn , µ∈(−0,n) and mq+µ>n, such that
mn+m n
−
|a(x,z,ξ)|≤Λ ϕ(x)+|z|ℓ(mm−1) +|ξ|m−1 ,
(2.2)
|b(x,z,ξ)|≤Λ (cid:16)ψ(x)+|z|ℓ−1+|ξ|m(ℓℓ−1) (cid:17)
for a.a. x∈Ω and all (z,ξ)∈R×(cid:16)Rn. Here, ℓ is the Sobolev(cid:17)conjugate of m and is

given by
nm if m<n,
(2.3) ℓ= n m
−
(any exponent ℓ>n if m=n.
• Coercivity condition: There exists a constant γ >0 such that
(2.4) a(x,z,ξ)·ξ ≥γ|ξ|m−Λ|z|ℓ−Λϕ(x)mm−1
for a.a. x∈Ω and all (z,ξ)∈R×Rn.
Recall that a function u ∈ W1,m(Ω) is called weak solution to the Dirichlet
0
problem (1.1) if
(2.5) a(x,u(x),Du(x))·Dv(x)dx+ b(x,u(x),Du(x))v(x) dx=0
ZΩ ZΩ
for each test function v ∈W1,m(Ω). It is worth noting that the convergence of the
0
integrals involved in (2.5) for all admissible u and v is ensured by (2.2) under the
sole assumptions p≥ m and q ≥ mn when m<n, q >1 if m=n.
m 1 mn+m n
Throughout the pape−r the omnibus ph−rase “known quantities” means that a
given constant depends on the data in hypotheses (2.1)–(2.4), which include n, m,
ℓ,γ, Λ, p,λ, q, µ, kϕk , kψk , P, diamΩ, A and r . We will denote by
Lp,λ(Ω) Lq,µ(Ω) Ω 0
C a generic constant, depending on known quantities, which may vary within the
same formula.
Our first result claims global essential boundedness of the weak solutions to the
problem (1.1).
Theorem2.1. LetΩsatisfy (2.1)andassume (2.2)and (2.4). TheneachW1,m(Ω)-
0
weak solution to the problem (1.1) is globally essentially bounded. That is, there
exists a constant M, depending on known quantities, on kDuk and on the
Lm(Ω)
uniform integrability of |Du|m, such that
(2.6) kuk ≤M.
L∞(Ω)
QUASILINEAR EQUATIONS WITH MORREY DATA 7
An immediate consequence of Theorem 2.1 and the local properties of solutions
to quasilinear elliptic equations (cf. [17, 27]) is the interior Ho¨lder continuity of
the weak solutions.
Corollary 2.2. Under the hypotheses of Theorem 2.1, each weak solution to (1.1)
is locally Ho¨lder continuous in Ω. That is,
|u(x)−u(y)|
sup ≤H ∀Ω′ ⋐Ω
|x−y|α
x,y Ω′, x=y
∈ 6
with an exponent α∈(0,1)and a constant H >0 depending on the samequantities
as M in (2.6) and on dist(Ω,∂Ω) in addition.
′
Whatreallyturnsoutisthatassumptions(2.1),(2.2)and(2.4)arealsosufficient
to ensure Ho¨lder continuity of the weak solutions up to the boundary, and this is
the essence of our second main result.
Theorem 2.3. Assume (2.1), (2.2) and (2.4). Then each weak solution of the
Dirichlet problem (1.1) is globally Ho¨lder continuous in Ω. Precisely,
|u(x)−u(y)|
sup ≤H,
|x−y|α
x,y Ω, x=y
∈ 6
where the exponent α ∈ (0,1) and the Ho¨lder constant H > 0 depend on the same
quantities as M in (2.6).
3. Auxiliary Results
For the sake of completeness, we collect here some auxiliary results to be used
in proving Theorems 2.1 and 2.3.
3.1. Basic tools.
Proposition 3.1. (Embeddings between Morrey spaces, see [22]) For arbitrary
s,s ∈[1,∞) and θ ,θ ∈[0,n), one has
′ ′′ ′ ′′
Ls′,θ′(Ω)⊆Ls′′,θ′′(Ω)
if and only if
s s
′ ′′
s′ ≥s′′ ≥1 and ≥ .
n−θ n−θ
′ ′′
Proposition3.2. (Hartman–Stampacchiamaximumprinciple,see[10],[14]*ChapterII,
Lemma5.1)Let τ: R→[0,∞)beanon-increasing function andsuppose thereexist
constants C >0, k ≥0, δ >0 and α∈[0,1+δ] such that
0
∞τ(t)dt≤Ckα τ(k) 1+δ ∀k ≥k .
0
Zk
Then τ supports the finitetimeextinction(cid:0)prop(cid:1)erty, that is, there is a numberk ,
max
depending on C, k0, δ, α and k∞0 τ(t)dt, such that
Rτ(k)=0 ∀k ≥kmax.
Proposition 3.3. (Adams trace inequality, see [1]) Let M be a positive Radon
measure supported in Ω and such that M(Bρ(x)) ≤ Kρα0 for each x ∈ Rn and
each ρ>0, where K is an absolute constant and
s
α = (n−r), 1<r <s<∞, r <n.
0
r
8 S.-S.BYUN,D.K.PALAGACHEV,P.SHIN
Then
1/s 1/r
|v(x)|s dM ≤C(n,s,r)K1/s |Dv(x)|r dx ∀v ∈W1,r(Ω).
0
(cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
In particular, if dM=c(x)dx with c∈L1,n−r+ε0(Ω) and ε0 >0, then
1/s 1/r
|v(x)|sc(x)dx ≤C n,s,r,kck |Dv(x)|r dx
L1,n−r+ε0(Ω)
(cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
(cid:0) (cid:1)
for all v ∈W1,r(Ω), where n−r+ε = s(n−r), 1<r <s<∞, r <n.
0 0 r
Proposition 3.4. (Gehring–Giaquinta–Modica lemma, see [7, Proposition 1.1,
Chapter V]) Let B be a fixed ball and G ∈ Ls(B), F ∈ Ls0(B) be nonnegative
functions with s >s>1. Suppose
0
s
− Gs(x)dx≤c − G(x)dx +− Fs(x)dx+θ− Gs(x)dx
ZBρ ZB2ρ ! ZB2ρ ZB2ρ
for each ball B of radius ρ∈(0,ρ ) such that B ⊂B, where 0≤θ <1.
ρ 0 2ρ
Then there exist constants C and m ∈ (s,s ], depending on n, c, s, s and θ,
0 0 0
such that
1/m0 1/s 1/s0
− Gm0(x)dx ≤C − Gs(x)dx + − Fs0(x)dx .
ZBρ !  ZB2ρ ! ZB2ρ ! 
 
Proposition3.5. (John–Nirenberglemma,see[26,Lemma1.2],[8,Theorem7.21])
Let B be a ball in Rn, u∈W1,m(B ) and suppose that, for any ball B ⊂B with
0 0 0
the same center as B there exists a constant K such that
0
n−m
kDukLm(B) ≤K|B| mn .
Then there exists constants σ >0 and C depending on K,m,n such that
0
eσ0u dx e−σ0u dx≤C|B0|2.
ZB0 ZB0
Proposition 3.6. (see [8, Lemma 8.23]) Let F and G be nondecreasing functions
in an interval (0,R]. Suppose that for all ρ≤R one has
G(ρ/2)≤c G(ρ)+F(ρ)
0
for some 0<c0 <1. Then for any 0<τ(cid:0)<1 and ρ≤(cid:1)R we have
ρ α
G(ρ)≤C G(R)+F(ρτR1 τ)
−
R
(cid:16)(cid:16) (cid:17) (cid:17)
where C =C(c ) and α=α(c ,τ) are positive constants.
0 0
3.2. Boundary Sobolev inequality. The next result is a boundary variant of
the Sobolev inequality which holds under the P-thickness condition.
Lemma 3.7. (BoundarySobolevinequality) Let Ω be a bounded domain with uni-
formly P-thick complement Rn\Ω and consider a function u∈W1,m(Ω) which is
0
extended as zero outside Ω.
Let B beaball of radius ρ∈(0,r /(1−θ)), centeredat a point of Ω and suppose
ρ 0
B \Ω6=∅ for some 0<θ <1.
θρ
QUASILINEAR EQUATIONS WITH MORREY DATA 9
Then for any s∈[P,m] there is a constant C =C(n,θ,s,P,A ) such that
Ω
1/s˜ 1/s
(3.1) − |u(x)|s˜dx ≤Cρ − |Du(x)|s dx
ZBρ ! ZBρ !
for each s˜∈[s,s ], where s is the Sobolev conjugate of s (s =ns/(n−s) if s<n
∗ ∗ ∗
and s is any exponent greater than n otherwise).
∗
Proof. Without loss of generality we may suppose that u is an s-quasicontinuous
function in W1,s(B ). Since B \Ω 6= ∅, we can take a ball B (x ) of radius
ρ θρ (1 θ)ρ 0
−
(1−θ)ρ, centered at x ∈ ∂Ω and such that B (x ) ⊂ B . Setting N(u) =
0 (1 θ)ρ 0 ρ
−
{x ∈ B : u(x) = 0} and applying the Ho¨lder inequality and [11, Lemma 3.1], we
ρ
get
1/s˜ 1/s∗
(3.2) − |u(x)|s˜dx ≤ − |u(x)|s∗ dx
ZBρ ! ZBρ !
1/s
1
≤ C |Du(x)|s dx
Caps N(u),B2ρ ZBρ !
(cid:0) (cid:1)
whenever s<n. Indeed, (3.2) holds also for any s >n when s=n. In fact,
∗
Caps′ N(u),B2ρ ≤Cρn−s′Capn N(u),B2ρ s′/n
(cid:0) (cid:1) (cid:0) (cid:1)
for abtrary s <n, whence we have
′
1/s′ 1/n
1 1
|Du|s′ dx ≤ |Du|n dx .
Caps′ N(u),B2ρ ZBρ ! Capn N(u),B2ρ ZBρ !
(cid:0) (cid:1) (cid:0) (cid:1)
Takings = ns∗ <nin(3.2)andusingtheaboveinequality,weget(3.2)fors=n
′ n+s∗
and for arbitrary s >n.
∗
Since u = 0 in Rn\Ω except of a set of s-capacity zero and B (x ) ⊂ B ,
(1 θ)ρ 0 ρ
we have −
Cap N(u),B ≥Cap B \Ω,B ≥Cap B (x )\Ω,B
s 2ρ s ρ 2ρ s (1 θ)ρ 0 2ρ
−
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
by the properties of capacity, whereas
(3.3) Cap B (x )\Ω,B ≥C(n,θ,s)Cap B (x )\Ω,B (x ) .
s (1 θ)ρ 0 2ρ s (1 θ)ρ 0 2(1 θ)ρ 0
− − −
(cid:0) (cid:1) (cid:0) (cid:1)
In fact, to see (3.3) we take functions v ∈ C (B ), 0 ≤ v ≤ 1, v = 1 on
0∞ 2ρ
B (x )\Ω and η ∈ C (B (x )), |Dη| ≤ c , 0 ≤ η ≤ 1, η = 1 on
(1 θ)ρ 0 0∞ 2(1 θ)ρ 0 (1 θ)ρ
− − −
B (x ). Then vη ∈C (B (x )), 0≤ vη ≤1, vη =1 on B (x )\Ω
(1 θ)ρ 0 0∞ 2(1 θ)ρ 0 (1 θ)ρ 0
− − −
10 S.-S.BYUN,D.K.PALAGACHEV,P.SHIN
and therefore, if s<n, we have
(3.4) Cap B (x )\Ω,B (x )
s (1 θ)ρ 0 2(1 θ)ρ 0
− −
(cid:0)≤ |D(vη)|s dx (cid:1)
ZB2(1−θ)ρ(x0)
1 s
≤C |Dv|s dx+ |v|s dx
(1−θ)ρ)
ZB2(1−θ)ρ(x0) (cid:18) (cid:19) ZB2(1−θ)ρ(x0) !
s/s∗
≤C |Dv|s dx+ |v|s∗dx
ZB2ρ ZB2ρ ! 
 
≤C |Dv|s dx.
ZB2ρ
The same bound holds true also if s = n with a constant C, depending on θ in
addition. Actually, making use of the Ho¨lder and Sobolev ([9, 15.30]) inequalities,
for arbitrary t>1 we get
n n 1/t
1 1
|v|n dx≤ |v|nt dx |B |1 1/t
2ρ −
(1−θ)ρ) (1−θ)ρ)
(cid:18) (cid:19) ZB2(1−θ)ρ(x0) (cid:18) (cid:19) ZB2ρ !
1/t 1/t
1 1
≤ ρ−nt |v|nt dx = − |v|nt dx
(1−θ)n (1−θ)n
ZB2ρ ! ZB2ρ !
C C
≤ ρn− |Dv|n dx= |Dv|n dx
(1−θ)n (1−θ)n
ZB2ρ ZB2ρ
and thus (3.4) with s=n.
This way, (3.3) follows after taking the infimum in the right-hand side of (3.4)
over all v ∈C (B ) such that v =1 in B (x )\Ω.
0∞ 2ρ (1 θ)ρ 0
−
Further on, the uniform s-thickness condition (2.1) yields
Cap B (x )\Ω,B (x )
s (1 θ)ρ 0 2(1 θ)ρ 0
− −
(cid:0)≥C(n,s,P,AΩ)Caps B(1 θ)ρ((cid:1)x0),B2(1 θ)ρ(x0)
− −
=C(n,θ,s,P,AΩ)ρn−(cid:0)s (cid:1)
and therefore the desired estimate (3.1) follows from (3.3) and (3.2). (cid:3)
3.3. Higher integrability of the gradient. The next result provides a crucial
step to obtain global boundedness of the weak solutions to (1.1) although it is
interesting by its own. Actually, it shows that the gradient of the weak solution
to controlled growths and coercive problems (1.1) gains better integrability over
domains with P-thick complements.
Lemma 3.8. Assume (2.1), (2.2) and (2.4), and let u ∈ W1,m(Ω) be a weak
0
solution to the Dirichlet problem (1.1).
Then there exist exponents m0 >m and ℓ0 >ℓ such that u∈W1,m0(Ω)∩Lℓ0(Ω)
and
(3.5) kDuk +kuk ≤C
Lm0(Ω) Lℓ0(Ω)
with a constant C depending on known quantities, on kDuk and on the uni-
Lm(Ω)
form integrability of |Du|m in Ω.