Spectral estimates of the $p$-Laplace Neumann operator in conformal regular domains

0.2 MB·

by  Vladimir Gol'dshtein

SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN
OPERATOR IN CONFORMAL REGULAR DOMAINS
6
1
V.GOL’DSHTEINANDA.UKHLOV
0
2
Abstract. In this paper we study spectral estimates of the p-Laplace Neu-
n mann operator in conformal regular domains Ω ⊂ R2. This study is based
a
on (weighted) Poincaré-Sobolev inequalities. The main technical tool is the
J
composition operators theory in relation with the Brennan’s conjecture. We
7 prove that if the Brennan’s conjecture holds then for any p ∈ (4/3,2) and
1 r ∈ (1,p/(2−p)) the weighted (r,p)-Poincare-Sobolev inequality holds with

theconstantdependingontheconformalgeometryofΩ. Asaconsequencewe
]
P obtain classical Poincare-Sobolev inequalities and spectral estimates for the
first nontrivial eigenvalue of the p-Laplace Neumann operator for conformal
A
regular domains.
.
h
t
a
m
1. Introduction and methodology
[
Let Ω ⊂ R2 be a simply connected planar domain with a smooth boundary

2
∂Ω. We consider the Neumann eigenvalue problem for the p-Laplace operator
v
(1<p<2):
3
3 −div |∇u|p−2∇u =µ |u|p−2u in Ω
5 (1.1) p
1 (∂∂nu =(cid:0)0 (cid:1) on ∂Ω.
0
The weak statement of this spectral problem is as follows: a function u solves
.
1 the previous problem iff u∈W1,p(Ω) and
0
6 |∇u(x,y)|p−2∇u(x,y) ·∇v(x,y) dxdy =µ |u|p−2u(x,y)v(x,y) dxdy
1 ¨ p¨
: Ω (cid:0) (cid:1) Ω
v
i for all v ∈W1,p(Ω).
X
The first nontrivial Neumann eigenvalue µ can be characterizedas
p
r
a |∇u(x,y)|p dxdy
µ (Ω)=min ˜Ω :u∈W1,p(Ω)\{0}, |u|p−2u dxdy =0 .
p  |u(x,y)|p dxdy ¨ 
 ˜Ω Ω 
Moreover, µp(Ω)−p1 is the best constant Bp,p(Ω) (see, for example, [1]) inthe
following Poincaré-Sobolev inequality
inf kf −c|Lp(Ω)k≤B (Ω)k∇f |Lp(Ω)k, f ∈W1,p(Ω).
p,p
c∈R
We prove, that µ (Ω) depends on the conformal geometry of Ω and can be
p
estimated in terms of Sobolev norms of a conformal mapping of the unit disc D
onto Ω (Theorem A).
0
Key words and phrases: conformal mappings,Sobolevspaces,ellipticequations.
0
2010 Mathematics Subject Classification: 35P15,46E35, 30C65.
1
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 2
The main technical tool is existence of universal weighted Poincaré-Sobolev in-
equalities
1
r
(1.2) inf |f(x,y)−c|rh(x,y)dxdy
c∈R ¨
(cid:18) (cid:19)
Ω
1
p
≤B (Ω,h) |∇f(x,y)|p dxdy , f ∈W1,p(Ω),
r,p ¨
(cid:18) (cid:19)
Ω
inanysimplyconnecteddomainΩ6=R2forconformalweightsh(x,y):=J (x,y)=
ϕ
|ϕ′(x,y)|2 induced by conformal homeomorphisms ϕ:Ω→D.
Main results of this article can be divided onto two parts. The
first part is the technical one and concerns to weighted Poincaré-
Sobolev inequalities in arbitrary simply connected planar domains
with nonempty boundaries (Theorem C and its consequences). Results
ofthefirstpartwillbeusedfor(nonweighted)Poincaré-Sobolevin-
equalities in so-called conformal regular domains (Theorem B) that
leads to lower estimates for the first nontrivial eigenvalue µ (The-
p
orem A). To the best of our knowledge lower estimates were known
before for convex domains only. The class of conformal regular do-
mains is much larger. It includes, for example, bounded domains with
Lipschitzboundariesandquasidiscs, i.eimagesofdiscsunder quasicon-
formal homeomorphisms of whole plane.
Brennan’s conjecture [4] is that for a conformal mapping ϕ:Ω→D
4
(1.3) |ϕ′(x,y)|β dxdy <+∞, for all <β <4.
ˆ 3
Ω
For the inverse conformal mapping ψ = ϕ−1 : D → Ω Brennan’s conjecture [4]
states
2
(1.4) |ψ′(u,v)|α dudv <+∞, for all −2<α< .
¨ 3
D
AconnectionbetweenBrennan’sConjectureandcompositionoperatorsonSobolev
spaces was established in [14]:
Equivalence Theorem. Brennan’s Conjecture (1.3) holds for a number s ∈
(4/3;4) if and only if a conformal mapping ϕ:Ω→D induces a bounded composi-
tion operator
ϕ∗ :L1,p(D)→L1,q(p,β)(Ω)
for any p∈(2;+∞) and q(p,β)=pβ/(p+β−2).
The inverse Brennan’s Conjecture states that for any conformal mapping ψ :
D→Ω, the derivative ψ′ belongs to the Lebesgue space Lα(D), for −2<α<2/3.
The integrability of the derivative in the power greater then 2/3 requires some
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 3
restrictions on the geometry of Ω. If Ω ⊂R2 is a simply connected planar domain
of finite area, then
|ψ′(u,v)|2 dudv = J (u,v) dudv =|Ω|<∞.
¨ ¨ ψ
D D
Integrability of the derivative in the power α > 2 is impossible without additional
assumptions on the geometry of Ω. For example, for any α > 2 the domain Ω
necessarily has a finite geodesic diameter [15].
Let Ω⊂R2 be a simply connected planar domain. Then Ω is called a conformal
α-regular domain if there exists a conformal mapping ϕ:Ω→D such that
(1.5) (ϕ−1)′(u,v) α dudv <∞ for some α>2.
¨
D (cid:12) (cid:12)
(cid:12) (cid:12)
If Ω is a conformal α-regular domain for some α>2 we call Ω a conformal regular
domain.
Thepropertyofα-regularitydoesnotdependsonchoiceofaconformalmapping
ϕ and depends on the hyperbolic geometry of Ω only. For connection between
conformal mapping and hyperbolic geometry see, for example, [2].
Notethataboundary∂ΩofaconformalregulardomaincanhaveanyHausdorff
dimension between one and two, but can not be equal two [18].
Thenexttheoremgiveslowerestimatesofthefirstnontrivialp-LaplaceNeumann
eigenvalue:
Theorem A. Let ϕ:Ω→D be a conformal homeomorphism from a conformal
α-regular domain Ω to the unit disc D and Brennan’s Conjecture holds. Then for
every p∈(max{4/3,(α+2)/α},2) the following inequality is correct
1
≤
µ (Ω)
p
p−q
q
q∈[1,2ipn/f(4−p))k(ϕ−1)′|Lα(D)k2¨D | ϕ−1 ′|(pp−−2q)q dudv ·Bpαα−p2,q(D).
(cid:0) (cid:1)
 
Here B (D) isthe exact constant in the corresponding (r,q)-Poincare-Sobolev in-
r,q
equality in the unit disc D for r =αp/(α−2).
In the limit case α=∞ and p=q we have
Corollary A.Suppose that Ω is smoothly bounded Jordan domain with a bound-
ary ∂Ω of a class C1 with a Dini continuous normal. Let ϕ:Ω→D be a conformal
homeomorphismfromΩontotheunitdiscD. Thenforeveryp∈(1,2)thefollowing
inequality is correct
1 1
≤k(ϕ−1)′|L∞(D)kp .
µ (Ω) µ (D)
p p
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 4
Remark 1.1. The constant B (D) satisfies [9, 16]:
r,q
1−δ
2 1−δ
B (D)≤ , δ =1/q−1/r.
r,q πδ 1/2−δ
(cid:18) (cid:19)
Remark 1.2. The Brennan’s conjecture was proved for α ∈ [α ,2/3) when α =
0 0
−1.752 [17].
Remark 1.3. The numeric estimates for the µ (Ω) were known before only for
p
convex domains. For example, in [8] was proved that
p
π
p
µ (Ω)≥
p
d(Ω)
(cid:18) (cid:19)
where
1
(p−1)p
1
dt (p−1)p
π =2 =2π .
p ˆ (1−tp/(p−1))p1 p(sin(π/p))
0
Remark1.4. Theorem Ahasadirectconnectionwiththespectralstabilityproblem
for the p-Laplace operator. See, the recent papers, [6, 7, 20], where one can found
the history of the problem, main results in this area and appropriate references.
Theorem A is a corollary (after simple calculations) of the following version of
the Poincaré-Sobolev inequality
Theorem B. Suppose that Ω⊂R2 be a conformal α-regular domain and Bren-
nan’s Conjecture holds. Then for every p ∈ (max{4/3,α/(α−1)},2), every s ∈
(1,α−2 p ) and every function f ∈W1,p(Ω), the inequality
α 2−p
1 1
s p
(1.6) inf |f(x,y)−c|sdxdy ≤B (Ω) |∇f(x,y)|p dxdy
c∈R ¨ s,p ¨
(cid:18) (cid:19) (cid:18) (cid:19)
Ω Ω
holds with the constant
Bs,p(Ω)≤k(ϕ−1)′|Lα(D)k2sBr,p(Ω,h)
≤q∈[1,2ipn/f(4−p)) Bαα−s2,q(D)·k(ϕ−1)′|Lα(D)k2sKp,q(D) .
n o
Here B (Ω,h), r = αs/(α−2), is the best constant of the following weighted
r,p
Poincaré-Sobolev inequality:
Theorem C. Suppose Ω ⊂ R2 is a simply connected domain with non empty
boundary, Brennan’s Conjecture holds and h(z) = J(z,ϕ) is the conformal weight
defined by a conformal homeomorphism ϕ : Ω → D. Then for every p ∈ (4/3,2)
and every function f ∈W1,p(Ω), the inequality
1 1
r p
(1.7) inf |f(x,y)−c|rh(x,y)dxdy ≤B (Ω,h) |∇f(x,y)|p dxdy
c∈R ¨ r,p ¨
(cid:18) (cid:19) (cid:18) (cid:19)
Ω Ω
holds for any r ∈[1,p/(2−p)) with the constant
B (Ω,h)≤ inf {K (D)·B (D)}
r,p p,q r,q
q∈[1,2p/(4−p))
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 5
.
Here B (D) is the best constant in the (non-weighted) (r.q)-Poincaré-Sobolev
r,q
inequalityintheunitdiscD⊂R2 andK (Ω)isthenormofcompositionoperator
p,q
ϕ−1 ∗ :L1,p(Ω)→L1,q(D)
generated by the inverse con(cid:0)form(cid:1)al mapping ϕ−1 :D→Ω:
p−q
pq
Kp,q(Ω)≤¨ | ϕ−1 ′|(pp−−2q)q dudv .
D
(cid:0) (cid:1)
 
Remark 1.5. Theorem C is correct (without referring of Brennan’s conjecture) for
2p |α | p
0
(1.8) 1≤r ≤ · <
2−p 2+|α | 2−p
0
and p∈((|α |+2)/(|α |+1),2), where α =−1.752 represents the best result for
0 0 0
which Brennan’s conjecture was proved.
Remark 1.6. Let Ω ⊂ R2 be a simply connected smooth domain. Then ϕ−1 ∈
Lα(D) for all α∈R and we have the weighted Poincaré-Sobolev inequality (1.2) for
all p∈[1,2) and all r∈[1,2p/(2−p)].
In the case, when we have embedding of weighted Lebesgue spaces in non-
weighted one, the weighted Poincaré-Sobolev inequality (1.7) implies the standard
Poincaré-Sobolev inequality (1.6).
Let us give some historical remarks about the notion of conformal regular do-
mains. This notion was introduced in [5] and was applied to the stability problem
for eigenvalues of the Dirichlet-Laplace operator. In [16] the lower estimates for
the first non-trivialeigenvalues of the the Neumann-Laplace operatorin conformal
regular domains were obtained. In [15] we proved but did not formulated the fol-
lowing important fact about conformal regular domains and the Poincaré-Sobolev
inequality:
Theorem 1.7. Let a simply connected domain Ω ⊂ R2 of finite area does not
support the (s,2)-Poincaré-Sobolev inequality
1 1
s 2
inf |f(x,y)−c|s dxdy ≤B (Ω) |∇f(x,y)|2 dxdy
c∈R ¨ s,2 ¨
(cid:18) (cid:19) (cid:18) (cid:19)
Ω Ω
for some s≥2. Then Ω is not a conformal regular domain.
Inthepresentworkwesuggestforthe conformalregulardomainsanewmethod
basedonthe compositionoperatorstheory. The suggestedmethod ofinvestigation
is based on the composition operators theory [23, 24] and its applications to the
Sobolev type embedding theorems [10, 11].
The following diagram illustrate this idea:
W1,p(Ω) −ϕ→∗ W1,q(D)
↓ ↓
Ls(Ω) (ϕ←−−1)∗ Lr(D)
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 6
Heretheoperatorϕ∗ definedbythecompositionruleϕ∗(f)=f◦ϕisabounded
composition operator on Sobolev spaces induced by a homeomorphism ϕ of Ω and
D and the operator (ϕ−1)∗ defined by the composition rule (ϕ−1)∗(f) = f ◦ϕ−1
is a bounded composition operator on Lebesgue spaces. This method allows to
transferPoincaré-Sobolevinequalitiesfromregulardomains(forexample,fromthe
unit disc D) to Ω.
In the recent works we studied compositionoperators on Sobolev spaces in con-
nectionwiththeconformalmappingstheory[12]. Thisconnectionleadstoweighted
Sobolevembeddings[13,14]withtheuniversalconformalweights. Anotherapplica-
tionofconformalcompositionoperatorswasgivenin[5]wherethespectralstability
problem for conformal regular domains was considered.
2. Composition Operators
Becauseallcompositionoperatorsthatwillbe usedinthis paperareinducedby
conformal homeomorphisms we formulate results about composition operators for
diffeomorphisms only.
2.1. Composition Operators on Lebesgue Spaces. For any domain Ω ⊂ R2
and any 1≤p<∞ we consider the Lebesgue space
1/p
Lp(Ω):= f :Ω→R:kf |Lp(Ω)k:= |f(x,y)|p dxdy <∞ .
 ¨  
 Ω 
 
The following theorem about composition operators on Lebesgue spacesis well
known (see, for example [24]):
Theorem 2.1. Let ϕ :Ω → Ω′ be a diffeomorphism between two planar domains
Ω and Ω′. Then the composition operator
ϕ∗ :Lr(Ω′)→Ls(Ω), 1≤s≤r <∞,
is bounded, if and only if and
r−s
r rs
¨ Jϕ−1(u,v) r−s dudv =K <∞, 1≤s<r<∞,
(cid:18)Ω′ (cid:0) (cid:1) (cid:19)
1
sup Jϕ−1(u,v) s =K <∞, 1≤s=r<∞.
(u,v)∈Ω′
(cid:0) (cid:1)
The norm of the composition operator kϕ∗k=K.
2.2. CompositionOperatorsonSobolevSpaces. WedefinetheSobolevspace
W1,p(Ω), 1 ≤ p < ∞ as a Banach space of locally integrable weakly differentiable
functions f :Ω→R equipped with the following norm:
1 1
p p
kf |W1,p(Ω)k= |f(x,y)|pdxdy + |∇f(x,y)|pdxdy .
¨ ¨
(cid:18) (cid:19) (cid:18) (cid:19)
Ω Ω
We define also the homogeneous seminormed Sobolev space L1,p(Ω) of locally
integrable weakly differentiable functions f : Ω → R equipped with the following
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 7
seminorm:
1
p
kf |L1,p(Ω)k= |∇f(x,y)|pdxdy .
¨
(cid:18) (cid:19)
Ω
Recall that the embedding operator i:L1,p(Ω)→L1 (Ω) is bounded.
loc
Let Ω and Ω′ be domains in R2. We say that a diffeomorphism ϕ : Ω → Ω′
induces a bounded composition operator
ϕ∗ :L1,p(Ω′)→L1,q(Ω), 1≤q ≤p≤∞,
bythecompositionruleϕ∗(f)=f◦ϕifϕ∗(f)∈L1,q(Ω)andthereexistsaconstant
K <∞ such that
kϕ∗(f)|L1,q(Ω)k≤Kkf |L1,p(Ω′)k.
The main result of [23] gives the analytic description of composition operators
on Sobolev spaces (see, also [24]) and asserts (in the case of diffeomorphisms) that
Theorem 2.2. [23] A diffeomorphism ϕ:Ω→Ω′ between two domains Ω and Ω′
induces a bounded composition operator
ϕ∗ :L1,p(Ω′)→L1,q(Ω), 1≤q <p<∞,
if and only if
q p−q
|ϕ′(x,y)|p p−q pq
K (Ω)= dxdy <∞.
p,q ¨ |J (x,y)|
(cid:18) (cid:18) ϕ (cid:19) (cid:19)
Ω
Definition 2.3. We calla boundeddomain Ω⊂R2 as a(r,q)-embedding domain,
1≤q,r ≤∞, if the embedding operator
i :W1,q(Ω)֒→Lr(Ω)
Ω
is bounded. The unit disc D ⊂ R2 is an example of the (r,2)-embedding domain
for all r ≥1.
The following theorem gives a characterization of composition operators in the
normed Sobolev spaces[16]. For readers convenience we reproduce here the proof
of the theorem.
Theorem 2.4. Let Ω be an (r,q)-embedding domain for some 1 ≤ q ≤ r ≤ ∞
and |Ω′| < ∞. Suppose that a diffeomorphism ϕ : Ω → Ω′ induces a bounded
composition operator
ϕ∗ :L1,p(Ω′)→L1,q(Ω), 1≤q ≤p<∞,
and the inverse diffeomorphism ϕ−1 :Ω′ →Ω induces a bounded composition oper-
ator
(ϕ−1)∗ :Lr(Ω)→Ls(Ω′)
for some p≤s≤r.
Then ϕ:Ω→Ω′ induces a bounded composition operator
ϕ∗ :W1,p(Ω′)→W1,q(Ω), 1≤q ≤p<∞.
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 8
Proof. Because the composition operator (ϕ−1)∗ : Lr(Ω) → Ls(Ω′) is bounded,
then the following inequality
k(ϕ−1)∗g |Ls(Ω′)k≤A (Ω)kg |Lr(Ω)k
r,s
is correct. Here A (Ω) is a positive constant.
r,s
If a domain Ω is an embedding domain and the composition operators
(ϕ−1)∗ :Lr(Ω)→Ls(Ω′), ϕ∗ :L1,p(Ω′)→L1,q(Ω)
are bounded, then for a function f =g◦ϕ−1 the following inequalities
inf kf −c|Ls(Ω′)k≤A (Ω)inf kg−c|Lr(Ω)k
r,s
c∈R c∈R
≤A (Ω)Mkg |L1,q(Ω)k≤A (Ω)K (Ω)Mkf |L1,p(Ω′)k
r,s r,s p,q
hold. Here M and K (Ω) are positive constants.
p,q
The Hölder inequality implies the following estimate
|c|=|Ω′|−p1kc|Lp(Ω′)k≤|Ω′|−p1 kf |Lp(Ω′)k+kf −c|Lp(Ω′)k
≤|(cid:0)Ω′|−p1kf |Lp(Ω′)k+|Ω′|−1skf −(cid:1)c|Ls(Ω′)k.
Because q ≤r we have
kg |Lq(Ω)k≤kc|Lq(Ω)k+kg−c|Lq(Ω)k≤|c||Ω|q1 +|Ω|r−rqkg−c|Lr(Ω)k
≤ |Ω′|−p1kf |Lp(Ω′)k+|Ω′|−1skf −c|Ls(Ω′)k |Ω|q1 +|Ω|r−rqkg−c|Lr(Ω)k.
(cid:18) (cid:19)
From previous inequalities we obtain for ϕ∗(f)=g finally
kg |Lq(Ω)k≤|Ω|1q|Ω′|−p1kf |Lp(Ω′)k
+Ar,s(Ω)Kp,q(Ω)M|Ω|q1|Ω′|−p1kf |L1,p(Ω′)k
+Kp,q(Ω)M|Ω|r−rqkf |L1,p(Ω)k.
Therefore the composition operator
ϕ∗ :W1,p(Ω′)→W1,q(Ω)
is bounded. (cid:3)
3. Poincaré-Sobolev inequalities
3.1. WeightedPoincare-Sobolevinequalities. LetΩ⊂R2beaplanardomain
and let v :Ω →R be a smooth positive real valued function in Ω. For 1 ≤ p <∞
consider the weighted Lebesgue space
1/p
Lp(Ω,v):= f :Ω→R:kf |Lp(Ω,v)k:= |f(x,y)|pv(x,y) dxdy <∞ .
¨
( (cid:18) Ω (cid:19) )
It is a Banach space for the norm kf |Lp(Ω,v)k.
Applicationsoftheconformalmappingstheorytothe Poincaré-Sobolevinequal-
ities in planar domains is based on the following result (Theorem 3.3, Proposition
3.4 [14]) which connected the classical mappings theory and the Sobolev spaces
theory.
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 9
Theorem 3.1. Let Ω ⊂R2 be a simply connected domain with non-empty bound-
ary and ϕ : Ω → D be a conformal homeomorphism. Suppose that the (Inverse)
Brennan’s Conjecture holds for the interval [α ,2/3) where α ∈ − 2,0 and
0 0
p∈ |α0|+2,2 .
|α0|+1 (cid:0) (cid:1)
Then the inverse mapping ϕ−1 induces a bounded composition operator
(cid:0) (cid:1)
(ϕ−1)∗ :L1(Ω)→L1(D)
p q
for any q such that
p|α | 2p
0
1≤q ≤ <
2+|α |−p 4−p
0
and for any function g ∈L1(Ω) the inequality
p
p−q
pq
k(ϕ−1)∗f |L1,q(D)k≤ |(ϕ−1)′|(pp−−2q)q dudv kf |L1,p(Ω)k.
¨ 
D
 
Remark 3.2. Let us remark that |α0|+2 > 4 for any α ∈ −2,0 .
|α0|+1 3 0
Using this theorem we prove (cid:0) (cid:1)
Theorem C. Suppose that Ω ⊂ C is a simply connected domain with non
empty boundary, the Brennan’s Conjecture holds for the interval [α ,2/3), where
0
α ∈ −2,0 and h(z) = J (z) is the conformal weight defined by a conformal
0 ϕ
homeomorphism ϕ : Ω → D. Then for every p ∈ ((|α |+2)/(|α |+1),2) and
0 0
(cid:0) (cid:1)
every function f ∈W1,p(Ω), the inequality
1 1
r p
inf |f(x,y)−c|rh(z)dxdy ≤B (Ω,h) |∇f(x,y)|p dxdy
c∈R ¨ r,p ¨
(cid:18) (cid:19) (cid:18) (cid:19)
Ω Ω
holds for any r such that
2p |α | p
0
1≤r ≤ · <
2−p 2+|α | 2−p
0
with the constant
B (Ω,h)≤ inf {K (D)·B (D)}.
r,p p,q r,q
q∈[1,2p/(4−p))
Here B (D) is the best constant in the (non-weighted) Poincaré-Sobolev in-
r,q
equality in the unit disc D⊂C and K (Ω) is the norm of composition operator
p,q
ϕ−1 ∗ :L1,p(Ω)→L1,q(D)
generated by the inverse con(cid:0)form(cid:1)al mapping ϕ−1 :D→Ω:
p−q
pq
Kp,q(Ω)≤¨ | ϕ−1 ′|(pp−−2q)q dudv .
D
(cid:0) (cid:1)
 
SPECTRAL ESTIMATES OF THE p-LAPLACE NEUMANN OPERATOR 10
Proof. By the Riemann Mapping Theorem, there exists a conformal mapping ϕ :
Ω→D, and by the (Inverse) Brennan’s Conjecture,
|(ϕ−1)′(u,v)|α dudv <+∞, for all −2<α <α<2/3.
ˆ 0
D
Hence, by Theorem 3.1, the inequality
k∇(f ◦ϕ−1)|Lq(D)k≤K (D)k∇f |Lp(Ω)k
p.q
holds for every function f ∈L1,p(Ω) and for any q such that
p|α | 2p
0
(3.1) 1≤q ≤ < .
2+|α |−p 4−p
0
Choose arbitrarily f ∈ C1(Ω). Then g = f ◦ϕ−1 ∈ C1(D) and, by the classical
Poincaré-Sobolev inequality,
(3.2) inf kf ◦ϕ−1−c|Lr(D)k≤B (D)k∇(f ◦ϕ−1)|Lq(D)k
q,r
c∈R
for any r such that
2q
1≤r ≤
2−q
By elementary calculations from the inequality (3.1) follows
2q 2p |α | p
0
≤ · <
2−q 2−p 2+|α | 2−p
0
Combininginequalitiesfor2q/(2−q)andr weconcludethattheinequality(3.2)
holds for any r such that
2p |α | p
0
1≤r≤ · < .
2−p 2+|α | 2−p
0
Using the change of variable formula, the classical Poincaré-Sobolev inequality
for the unit disc
1
1 q
r
inf |g(u,v)−c|r dudv ≤B (D) |∇g(u,v)|q dudv
c∈R ¨ r,q ¨ 
(cid:18)D (cid:19) D
 
and Theorem 3.1, we finally infer
1
r
inf |f(x,y)−c|rh(x,y) dxdy =
c∈R ¨
(cid:18) (cid:19)
Ω
1 1
r r
inf |f(x,y)−c|rJ (x,y) dxdy = inf |g(u,v)−c|r dudv
c∈R ¨ ϕ c∈R ¨
(cid:18)Ω (cid:19) (cid:18)D (cid:19)
1 1
q p
≤B (D) |∇g(u,v)|q dudv ≤K (D)·B (D) |∇f(x,y)|p dxdy .
r,q ¨  p,q r,q ¨ 
D Ω
   