Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

0.26 MB·English

by  Daniele Castorina

Regularity of stable solutions of p-Laplace
equations through geometric Sobolev type
2 inequalities
1
0
2
Daniele Castorina Manel Sancho´n
n
a
J

7
1

]
P
A
Abstract
.
h
t
a InthispaperweproveaSobolevandaMorreytypeinequalityinvolvingthemean
m curvatureandthetangentialgradientwithrespecttothelevelsetsofthefunctionthat
[ appears in the inequalities. Then, as an application, we establish a priori estimates

1 forsemi-stablesolutions of ∆pu = g(u)inasmoothboundeddomainΩ Rn.In
− ⊂
v particular,weobtainnewLr andW1,r boundsfortheextremalsolutionu⋆whenthe
6 domain is strictly convex. More precisely, weprove that u⋆ L∞(Ω) ifn p+2
48 andu⋆ Ln−npp−2(Ω) W1,p(Ω)ifn > p+2. ∈ ≤
3 ∈ ∩ 0
. Keywords.Geometricinequalities,meancurvatureoflevelsets,Schwarzsymmetri-
1
0 zation,p-Laplaceequations, regularity ofstablesolutions
2
1
:
v 1 Introduction
i
X
r Theaimofthispaperistoobtainaprioriestimatesforsemi-stablesolutionsofp-Laplace
a
equations.Wewillaccomplishthisbyprovingsomegeometrictypeinequalitiesinvolving
thefunctionals
1/p
1 q
I (v;Ω) := v p/q + H q v pdx , p,q 1 (1.1)
p,q p′|∇T,v|∇ | | | v| |∇ | ≥
(cid:18)ZΩ(cid:16) (cid:17) (cid:19)
where Ω is a smooth bounded domain of Rn with n 2 and v C∞(Ω). Here, and in
≥ ∈ 0
therest ofthepaper,H (x) denotesthemean curvatureat xofthehypersurface y Ω :
v
{ ∈
v(y) = v(x) (whichissmoothatpointsx Ωsatisfying v(x) = 0),and isthe
T,v
| | | |} ∈ ∇ 6 ∇
D. Castorina: Departamentde Matema`tiques, UniversitatAuto`nomade Barcelona, 08193Bellaterra,
Spain;e-mail:[email protected]
M.Sancho´n:DepartamentdeMatema`ticaAplicadaiAna`lisi,UniversitatdeBarcelona,GranVia585,
08007Barcelona,Spain;e-mail:[email protected]
MathematicsSubjectClassification(2010):Primary35K57,35B65;Secondary35J60
1
2 DanieleCastorina,ManelSancho´n
tangentialgradientalongalevelsetof v .WewillproveaMorrey’stypeinequalitywhen
| |
n < p+q and aSobolevinequalitywhen n > p+q (see Theorem1.2 below).
Then, as an application of these inequalities, we establish Lr and W1,r a priori esti-
matesforsemi-stablesolutionsofthereaction-diffusionproblem
∆ u = g(u) inΩ,
p
−
u > 0 inΩ, (1.2)

u = 0 on∂Ω.

Here, the diffusion is modeled by the p-Laplace operator ∆ (remember that ∆ u :=
p p
div( u p−2 u)) with p > 1, while the reaction term is driven by any positive C1 non-
|∇ | ∇
linearityg.
As we will see, these estimates will lead to new Lr and W1,r bounds for the extremal
solution u⋆ of (1.2) when g(u) = λf(u) and the domain Ω is strictly convex. More pre-
cisely, we prove that u⋆ L∞(Ω) if n p + 2 and u⋆ Ln−npp−2(Ω) W1,p(Ω) if
∈ ≤ ∈ ∩ 0
n > p+2.
1.1 Geometric Sobolev inequalities
BeforeweestablishourSobolevand Morreytypeinequalitieswewillstatethatthefunc-
tionalI definedin(1.1)decreases(uptoauniversalmultiplicativeconstant)bySchwarz
p,q
symmetrization.GivenaLipschitzcontinuousfunctionv anditsSchwarzsymmetrization
v∗ itis wellknownthat
v∗ r dx = v r dx forall r [1,+ ]
| | | | ∈ ∞
ZBR ZΩ
and
v∗ r dx v r dx forallr [1, ).
|∇ | ≤ |∇ | ∈ ∞
ZBR ZΩ
OurfirstresultestablishesthatI (v∗;B ) CI (v;Ω)forsomeuniversalconstant
p,q R p,q
≤
C dependingonlyonn, p, and q.
Theorem 1.1. Let Ω be a smooth bounded domain of Rn with n 2 and B the ball
R
≥
centered at the origin and with radius R = ( Ω / B )1/n. Let v C∞(Ω) and v∗ its
| | | 1| ∈ 0
Schwarz symmetrization. Let I be the functional defined in (1.1) with p,q 1. If n >
p,q
≥
q +1 thenthereexistsa universalconstantC dependingonlyon n, p, andq, suchthat
1/p
1
v∗ pdx = I (v∗;B ) CI (v;Ω). (1.3)
x q|∇ | p,q R ≤ p,q
(cid:18)ZBR | | (cid:19)
NotethattheSchwarzsymmetrizationofvisaradialfunction,andhence,itslevelsets
are spheres. In particular, the mean curvature Hv∗(x) = 1/ x and the tangential gradient
| |
T,v∗ v∗ p/q = 0. Thisexplainstheequalityin(1.3).
∇ |∇ |
p-LaplaceequationsandgeometricSobolevinequalities 3
ArelatedresultwasprovedbyTrudinger[18]whenq = 1fortheclassofmeanconvex
functions (i.e., functions for which the mean curvature of the level sets is nonnegative).
Moreprecisely,heprovedTheorem 1.1replacing thefunctionalI by
p,q
1/p
I˜ (v;Ω) := H q v pdx (1.4)
p,q v
| | |∇ |
(cid:18)ZΩ (cid:19)
andconsideringtheSchwarzsymmetrizationofv withrespecttotheperimeterinsteadof
theclassicalonelikeus(seeDefinition2.1below).Inordertodefinethissymmetrization
(with respect to the perimeter) it is essential to know that the mean curvature H of the
v
level sets of v is nonnegative. Then using an Aleksandrov-Fenchel inequality for mean
| |
convex hypersurfaces (see [17]) he proved Theorem 1.1 for this class of functions when
q = 1.
WeproveTheorem1.1usingtwoingredients.Thefirstoneistheclassicalisoperimet-
ricinequality:
n B 1/n D (n−1)/n ∂D (1.5)
1
| | | | ≤ | |
foranysmoothboundeddomainDofRn.ThesecondoneisageometricSobolevinequal-
ity, due to Michael and Simon [12] and to Allard [1], on compact (n 1)-hypersurfaces
−
M withoutboundarywhichinvolvesthemeancurvatureH ofM:foreveryq [1,n 1),
∈ −
thereexistsaconstantAdependingonlyon nand q suchthat
1/q⋆ 1/q
φ q⋆dσ A φ q + Hφ q dσ (1.6)
| | ≤ |∇ | | |
(cid:18)ZM (cid:19) (cid:18)ZM (cid:19)
foreveryφ C∞(M),whereq⋆ = (n 1)q/(n 1 q)anddσdenotestheareaelementin
∈ − − −
M.Usingtheclassicalisoperimetricinequality(1.5)andthegeometricSobolevinequality
(1.6) with M = x Ω : v(x) = t and φ = v (p−1)/q we will prove Theorem 1.1
{ ∈ | |q } q |∇ |
withtheexplicitconstantC = Ap ∂B (n−1)p, beingAtheuniversalconstantin(1.6).
1
| |
From Theorem 1.1 and well known 1-dimensionalweighted Sobolev inequalitiesit is
easy to prove Morrey and Sobolev geometric inequalities involving the functional I .
p,q
Indeed, by Theorem 1.1 and since Schwarz symmetrization preserves the Lr norm, it is
sufficientto provetheexistenceofapositiveconstantC independentofv∗ such that
v∗ CI (v∗;B ).
k kLr(BR) ≤ p,q R
Usingthisargumentweprovethefollowinggeometricinequalities.
Theorem1.2. LetΩbeasmoothboundeddomainofRn withn 2andv C∞(Ω).Let
≥ ∈ 0
I bethefunctionaldefinedin (1.1)with p,q 1 and
p,q
≥
np
p⋆ := .
q n (p+q)
−
Assumen > q +1.The followingassertionshold:
4 DanieleCastorina,ManelSancho´n
(a) If n < p+q then
p+q−n
v L∞(Ω) C1 Ω np Ip,q(v;Ω) (1.7)
k k ≤ | |
forsomeconstantC dependingonlyonn, p, andq.
1
(b) If n > p+q, then
kvkLr(Ω) ≤ C2|Ω|r1−p1⋆qIp,q(v;Ω) for every1 ≤ r ≤ p⋆q, (1.8)
whereC isa constantdependingonlyonn, p, q, andr.
2
(c) If n = p+q, then
p′
v n
exp | | dx Ω , where p′ = p/(p 1), (1.9)
C I (v;Ω) ≤ n 1| | −
ZΩ ((cid:18) 3 p,q (cid:19) ) −
forsomepositiveconstantC dependingonlyon nand p.
3
Cabre´ and the second author [6] proved recently Theorem 1.2 under the assumption
q p using a different method (without the use of Schwarz symmetrization). More
≥
precisely, they proved the theorem replacing the functional I (v;Ω) by the one de-
p,q
fined in (1.4), I˜ (v;Ω). Therefore, our geometric inequalities are only new in the range
p,q
1 q < p.
≤
Open Problem 1. Is Theorem 1.2 true for the range 1 q < p and replacing the func-
≤
tionalI (v;Ω) bytheonedefined in(1.4), I˜ (v;Ω)?
p,q p,q
This question has a posive answer for the class of mean convex functions. Trudinger
[18] proved this result for this class of functions when q = 1 and can be easily extended
foreveryq 1.However,toourknowledge,forgeneral functions(withoutmeanconvex
≥
levelsets)itis an openproblem.
1.2 Regularity of semi-stable solutions
The second part of the paper deals with a priori estimates for semi-stable solutions of
problem (1.2). Remember that a regular solution u C1(Ω) of (1.2) is said to be semi-
∈ 0
stableifthesecond variationoftheassociated energy functionalat uis nonnegativedefi-
nite,i.e.,
2
u
u p−2 φ 2+(p 2) φ ∇ g′(u)φ2 dx 0 (1.10)
|∇ | |∇ | − ∇ · u − ≥
ZΩ ( (cid:18) |∇ |(cid:19) )
foreveryφ H ,whereH denotesthespaceofadmissiblefunctions(seeDefinition4.1
0 0
∈
below). The class of semi-stable solutions includes local minimizers of the energy func-
tionalas wellas minimaland extremalsolutionsof(1.2)when g(u) = λf(u).
p-LaplaceequationsandgeometricSobolevinequalities 5
Using an appropriatetest function in (1.10) we provethe followinga prioriestimates
forsemi-stablesolutions.Thisresultextendstheonesin[3]and[6]fortheLaplaciancase
(p = 2)duetoCabre´ and thesecondauthor.
Theorem 1.3. Let g be any C∞ function and Ω Rn any smooth bounded domain.
⊂
Let u C1(Ω) be a semi-stable solution of (1.2), i.e., a solution satisfying (1.10). The
∈ 0
followingassertionshold:
(a) Ifn p+2then thereexistsa constantC dependingonlyonnand psuchthat
≤
1/p
kukL∞(Ω) ≤ s+ sC2/p|Ω|p+n2p−n |∇u|p+2dx foralls > 0. (1.11)
(cid:18)Z{u≤s} (cid:19)
(b) If n > p+2 thenthereexists aconstantC dependingonlyon nandp suchthat
u s n−n(pp+2) dx n−(npp+2) C u p+2 dx 1/p (1.12)
| |− ≤ s2/p |∇ |
(cid:18)Z{u>s}(cid:16) (cid:17) (cid:19) (cid:18)Z{u≤s} (cid:19)
foralls > 0.Moreover,thereexistsaconstantC dependingonlyonn,p,andr suchthat
np
u r dx C Ω + u n−(p+2) dx+ g(u) L1(Ω) (1.13)
|∇ | ≤ | | | | k k
ZΩ (cid:18) ZΩ (cid:19)
forall1 r < r := np2 .
≤ 1 (1+p)n−p−2
To prove (1.11) and (1.12) we use the semi-stability condition (1.10) with the test
functionφ = u η to obtain
|∇ |
4 n 1
u p/2 2 + − H2 u p η2dx u p η 2dx (1.14)
p2|∇T,u|∇ | | p 1 u|∇ | ≤ |∇ | |∇ |
ZΩ(cid:18) − (cid:19) ZΩ
for every Lipschitz function η in Ω with η = 0. Then, taking η = T u = min s,u ,
∂Ω s
| { }
weobtain(1.11)and(1.12)whenn = p+2byusingtheMorreyandSobolevinequalities
6
established in Theorem 1.2 with q = 2. The critical case n = p+2 is more involved. In
order to get (1.11) in this case, we take another explicit test function η = η(u) in (1.14)
andusethegeometricSobolevinequality(1.6).Thegradientestimateestablishedin(1.13)
will follow by using a technique introduced by Be´nilan et al. [2] to get the regularity of
entropysolutionsforp-LaplaceequationswithL1 data(seeProposition4.2).
The rest of the introduction deals with the regularity of extremal solutions. Let us
recall theproblemandsomeknownresultsin thistopic.Consider
∆ u = λf(u) in Ω,
− p (1.15)
u = 0 on ∂Ω, λ
(cid:26)
whereλ isapositiveparameterand f isaC1 positiveincreasingfunctionsatisfying
f(t)
lim = + . (1.16)
t→+∞ tp−1 ∞
6 DanieleCastorina,ManelSancho´n
Cabre´ and the second author [5] proved the existence of an extremal parameter λ⋆
∈
(0, ) such that problem (1.15) admits a minimal regular solution u C1(Ω) for
∞ λ λ ∈ 0
λ (0,λ⋆) and admits no regularsolutionforλ > λ⋆. Moreover,every minimalsolution
∈
u isasemi-stableforλ (0,λ⋆).
λ
∈
FortheLaplacian case(p = 2),thelimitofminimalsolutions
u⋆ := lim u
λ
λ↑λ⋆
isa weak solutionoftheextremalproblem(1.15) and it isknownas extremalsolution.
λ⋆
Nedev [13] proved, in the case of convex nonlinearities, that u⋆ L∞(Ω) if n 3 and
∈ ≤
u⋆ Lr(Ω) for all 1 r < n/(n 4) if n 4. Recently, Cabre´ [3], Cabre´ and the
∈ ≤ − ≥
second author [6], and Nedev [14] proved, in the case of convex domains and general
nonlinearities,thatu⋆ L∞(Ω) ifn 4 and u⋆ Ln2−n4(Ω) H1(Ω) ifn 5.
∈ ≤ ∈ ∩ 0 ≥
For arbitrary p > 1 it is unknown if the limit of minimal solutions u⋆ is a (weak or
entropy) solution of (1.15) . In the affirmative case, it is called the extremal solution of
λ⋆
(1.15) . However, in [15] it is proved that the limit of minimal solutions u⋆ is a weak
λ⋆
solution(in the distributionalsense) of (1.15) wheneverp 2 and f satisfies theaddi-
λ⋆
≥
tionalcondition:
thereexistsT 0 such that(f(t) f(0))1/(p−1) isconvexforall t T. (1.17)
≥ − ≥
Moreover,
u⋆ L∞(Ω) ifn < p+p′
∈
and
n
u⋆ Lr(Ω), forall r < r˜ := (p 1) , ifn p+p′.
∈ 0 − n (p+p′) ≥
−
This extends previous results of Nedev [13] for the Laplacian case (p = 2) and convex
nonlinearities.
Our next result improves the Lq estimate in [13, 15] for strictly convex domains. We
also prove that u⋆ belongs to the energy class W1,p(Ω) independently of the dimension
0
extendingan unpublishedresultofNedev [14]forp = 2 toeveryp 2 (seealso[6]).
≥
Theorem 1.4. Let f be an increasingpositiveC1 function satisfying(1.16). Assumethat
Ω is a smooth strictly convex domain of Rn. Let u C1(Ω) be the minimal solution of
λ ∈ 0
(1.15) .There existsa constantC independentof λsuchthat:
λ
(a) If n ≤ p+2then kuλkL∞(Ω) ≤ Ckf(uλ)kL1/1((pΩ−)1).
(b) If n > p+2 then kuλkLn−npp−2(Ω) ≤ Ckf(uλ)kL1/1((pΩ−)1). Moreover kuλkW01,p(Ω) ≤ C′
whereC′ isa constantdependingonlyonn, p, Ω, f and f(u ) .
λ L1(Ω)
k k
Assume,inaddition,p 2 andthat (1.17)holds.Then
≥
p-LaplaceequationsandgeometricSobolevinequalities 7
(i) If n p+2then u⋆ L∞(Ω). In particular,u⋆ C1(Ω).
≤ ∈ ∈ 0
(ii) If n > p+2 thenu⋆ Ln−npp−2(Ω) W1,p(Ω).
∈ ∩ 0
Remark1.5. Iff(u )isboundedinL1(Ω)byaconstantindependentofλ,thenparts(a)
λ
and(b)willleadautomaticallytotheassertions(i)and(ii)statedinthetheorem(without
therequirementthatp 2and(1.17)holdtrue).However,aswesaidbefore,theestimate
≥
f(u⋆) L1(Ω) is unknown in the general case, i.e, for arbitrary positive and increasing
∈
nonlinearitiesf satisfying(1.16)andarbitrary p > 1.
Open Problem 2. Is it true that f(u⋆) L1(Ω) for arbitrary positive and increasing
∈
nonlinearitiesf satisfying(1.16)?
Under assumptions p 2 and (1.17) it is proved in [15] that f(u⋆) Lr(Ω) for all
≥ ∈
1 r < n/(n p′) when n p′ and f(u⋆) L∞(Ω) if n < p′. In particular, one
≤ − ≥ ∈
has f(u⋆) L1(Ω) independently of the dimension n and the parameter p > 1. As a
∈
consequence, assertions (i) and (ii) follow immediately from parts (a) and (b) of the
theorem.
ToprovetheLr aprioriestimatesstatedinpart(a)and(b)wemakethreesteps.First,
weusethestrictconvexityofthedomainΩtoprovethat
x Ω : dist(x,∂Ω) < ε x Ω : u (x) < s
λ
{ ∈ } ⊂ { ∈ }
forasuitables.Thisisdoneusingamovingplaneprocedureforp-Laplaceequations(see
Proposition 3.1 below). Then, we prove that the Morrey and Sobolev type inequalities
stated in Theorem 1.2 for smoothfunctions,also hold for regular solutionsof (1.2)when
1 q 2.Finally,takingatestfunctionη relatedtodist( ,∂Ω)in(1.14)andproceeding
≤ ≤ ·
as in theproof ofTheorem 1.3 we willobtain the Lr a prioriestimates establishedin the
theorem.
The energy estimate established in parts (ii) and (b) of Theorem 1.4 follows by ex-
tendingtheargumentsofNedev[14]fortheLaplaciancase(seealsoTheorem2.9in[6]).
First,usingaPohoˇzaevidentityweobtain
1
u p dx u p x ν dσ, for allp > 1 and λ (0,λ⋆), (1.18)
|∇ λ| ≤ p′ |∇ λ| · ∈
ZΩ Z∂Ω
where dσ denotes the area element in ∂Ω and ν is the outward unit normal to Ω. Then,
using the strict convexity of the domain (as in the Lr estimates) and standard regularity
estimates for ∆ u = λf(u (x)) in a neighborhood of the boundary, we are able to
p λ
−
control the right hand side of (1.18) by a constant whose dependence on λ is given by a
functionof f(u ) .
λ L1(Ω)
k k
8 DanieleCastorina,ManelSancho´n
Remark 1.6. Let uscompare ourregularityresultswiththesharp ones provedbyCabre´,
Capella, and thesecond authorin [4]when Ω istheunitballB ofRn. In theradial case,
1
theextremalsolutionu⋆ of(1.15) is boundedifthedimensionn < p+ 4p .Moreover,
λ⋆ p−1
ifn p+ 4p then u⋆ W1,r(B )forall 1 r < r¯ , where
≥ p−1 ∈ 0 1 ≤ 1
np
r¯ := .
1
n 2 n−1 2
− p−1 −
q
In particular,u⋆ Lr(B ) forall1 r < r¯ , where
1 0
∈ ≤
np
r¯ := .
0
n 2 n−1 p 2
− p−1 − −
q
Itcanbeshownthattheseregularityresultsaresharpbytakingtheexponentialandpower
nonlinearities.
Note that the Lr(Ω)-estimate established in Theorem 1.4 differs with the sharp expo-
nent r¯ defined aboveby the term 2 n−1. Moreover,observe that r¯ is larger than p and
0 p−1 1
tends to it as n goes to infinity. In pqarticular, the best expected regularity independent of
the dimension n for the extremal solution u⋆ is W1,p(Ω), which is the one we obtain in
0
Theorem1.4.
1.3 Outline of the paper
The paper is organized as follows. In section 2 we prove Theorem 1.1 and the geometric
type inequalities stated in Theorem 1.2. In section 3 we prove that Theorem 1.2 holds
for solutions of (1.2) when 1 q 2. Moreover we give boundary estimates when the
≤ ≤
domainisstrictlyconvex.In section 4, we present thesemi-stabilitycondition(1.10)and
the space of admissible functions H . The rest of the section deals with the regularity of
0
semi-stablesolutionsprovingTheorems1.3and 1.4.
2 Geometric Hardy-Sobolev type inequalities
In this section we prove Theorems 1.1 and 1.2. As we said in the introduction, the geo-
metricinequalities establishedin Theorem 1.2 are new for the range 1 q < p sincethe
≤
caseq pwas provedin[6]. However,wewillgivetheproofinall cases usingSchwarz
≥
symmetrization,givingan alternativeprooffortheknownrangeofparameters q p.
≥
We start recalling the definition of Schwarz symmetrizationof a compact set and of a
Lipschitzcontinuousfunction.
Definition 2.1. We definetheSchwarzsymmetrizationof a compactset D Rn as
⊂
B (0) withR = ( D / B )1/n if D = ,
D∗ := R | | | 1| 6 ∅
if D = .
(cid:26) ∅ ∅
p-LaplaceequationsandgeometricSobolevinequalities 9
Letv beaLipschitzcontinuousfunctionin ΩandΩ := x Ω : v(x) t . Wedefine
t
{ ∈ | | ≥ }
theSchwarzsymmetrizationof v as
v∗(x) := sup t R : x Ω∗ .
{ ∈ ∈ t}
Equivalently,we can definetheSchwarz symmetrizationofv as
v∗(x) = inf t 0 : V(t) < B x n ,
1
{ ≥ | || | }
whereV(t) := Ω = x Ω : v(x) > t denotesthedistributionfunctionofv.
t
| | |{ ∈ | | }|
The first ingredient in the proof of Theorem 1.1 is the isoperimetric inequality for
functionsv in W1,1(Ω):
0
d
n B 1/nV(t)(n−1)/n P(t) := v dx fora.e. t > 0, (2.1)
1
| | ≤ dt |∇ |
Z{|v|≤t}
where P(t) stands for the perimeter in the sense of De Giorgi (the total variation of the
characteristicfunctionof x Ω : v(x) > t ).
{ ∈ | | }
The second ingredient is the following Sobolev inequality on compact hypersurfaces
withoutboundarydueto Michaeland Simon[12]and toAllard[1].
Theorem 2.2 ([1, 12]). Let M Rn be a C∞ immersed (n 1)-dimensional compact
⊂ −
hypersurface without boundary and φ C∞(M). If q [1,n 1), then there exists a
∈ ∈ −
constantAdependingonlyonn andq suchthat
1/q⋆ 1/q
φ q⋆dσ A φ q + Hφ q dσ , (2.2)
| | ≤ |∇ | | |
(cid:18)ZM (cid:19) (cid:18)ZM (cid:19)
whereH isthemeancurvatureofM,dσ denotestheareaelementinM,andq⋆ = (n−1)q.
n−1−q
AswesaidintheintroductionitiswellknownthatSchwarzsymmetrizationpreserves
theLr-normanddecreasestheW1,r-norm.Letusprovethatitalsodecreases(uptoamul-
tiplicativeconstant) the functional I defined in (1.1) using the isoperimetric inequality
p,q
(2.1) and the geometric inequality(2.2) applied to M = M = x Ω : v(x) = t and
t
{ ∈ | | }
φ = v (p−1)/q.
|∇ |
Proofof Theorem 1.1. Let v C∞(Ω), p 1, and 1 q < n 1. By Sard’s theorem,
∈ 0 ≥ ≤ −
almost every t (0, v L∞(Ω)) is a regular value of v . By definition, if t is a regular
∈ k k | |
valueof v ,then v(x) > 0 for all x Ω such that v(x) = t. Therefore, M := x
t
| | |∇ | ∈ | | { ∈
Ω : v(x) = t is a C∞ immersed (n 1) dimensional compact hypersurface of Rn
| | } − −
without boundary for every regular value t . Applying inequality (2.2) to M = M and
t
φ = v (p−1)/q weobtain
|∇ |
q/q⋆
v (p−1)qq⋆ dσ Aq v p−q1 q + H q v p−1dσ (2.3)
T,v v
|∇ | ≤ ∇ |∇ | | | |∇ |
(cid:18)ZMt (cid:19) ZMt(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10 DanieleCastorina,ManelSancho´n
for a.e. t (0, v L∞(Ω)), where Hv denotes the mean curvature of Mt, dσ is the area
∈ k k
elementin M ,Ais theconstantin (2.2)whichdependsonlyon nand q, and
t
(n 1)q
q⋆ := − .
n 1 q
− −
RecallthatV(t),beinganonincreasingfunction,isdifferentiablealmosteverywhereand,
thankstothecoarea formulaand thatalmosteveryt (0, v L∞(Ω)) isaregularvalueof
∈ k k
v ,wehave
| |
1
V′(t) = dσ and P(t) = dσ fora.e. t (0, v L∞(Ω)).
− v ∈ k k
ZMt |∇ | ZMt
ThereforeapplyingJenseninequalityandthenusingtheisoperimetricinequality(2.1),
weobtain
v (p−1)qq⋆+1 dσ qq⋆ P(t)p−1+qq⋆ (A1V(t)n−n1)p−1+qq⋆ (2.4)
|∇ | v ≥ ( V′(t))p−1 ≥ ( V′(t))p−1
(cid:18)ZMt |∇ |(cid:19) − −
fora.e. t (0, v L∞(Ω)), whereA1 := n B1 1/n.
∈ k k | |
Note that for radial functions the inequalities in (2.4) are equalities. Therefore, since
the Schwarz symmetrization v∗ of v is a radial function and it satisfies (2.3), with an
equalityand withconstantA = ∂B −1/(n−1), weobtain
1
| |
q/q⋆
v∗ (p−1)qq⋆ dσ = ∂B1 −n−q1 Hv∗ q v∗ p−1dσ
|∇ | | | | | |∇ |
(cid:18)Z{|v∗|=t} (cid:19) Z{v∗=t} (2.5)
(A1V(t)n−n1)p−1+qq⋆
= .
( V′(t))p−1
−
for a.e. t (0, v L∞(Ω)). Here, we used that V(t) = v > t = v∗ > t for a.e.
∈ k k |{| | }| |{| | }|
t (0, v L∞(Ω)).
∈ k k
Therefore, from (2.3), (2.4), and (2.5), weobtain
∂B1 −n−q1 Hv∗ q v∗ p−1dσ Aq T,v v p−q1 q + Hv q v p−1dσ,
| | | | |∇ | ≤ ∇ |∇ | | | |∇ |
Z{v∗=t} ZMt(cid:12) (cid:12)
(cid:12) (cid:12)
for a.e. t (0, v L∞(Ω)). Integrating the prev(cid:12)ious inequali(cid:12)ty with respect to t on
∈ k k
(0, v L∞(Ω)) and using the coarea formula we obtain inequality (1.3), with the explicit
k k q q
constantC = Ap ∂B (n−1)p, provingtheresult.
1
| |
q q
Remark 2.3. We obtained the explicit admissible constant C = Ap ∂B (n−1)p in (1.3),
1
| |
whereAis theuniversalconstantappearing in(2.2).
We prove Theorem 1.2 using Theorem 1.1 and known results on one dimensional
weightedSobolevinequalities.