Green's function for second order parabolic systems with Neumann boundary condition

0.34 MB·English

by  Jongkeun Choi

Green’s function for second order parabolic
˚
systems with Neumann boundary condition
3
1
0
Jongkeun Choi: Seick Kim;
2

n
a
J
Abstract
8
We study the Neumann Green’s function for second order parabolic

] systems in divergence form with time-dependent measurable coefficients
P
in acylindrical domain Q“Ωˆp´8,8q, whereΩĂRn isan open con-
A
nected set suchthat amultiplicativeSobolev embeddinginequalityholds
h. there. Suchadomainincludes,forexample,aboundedSobolevextension
t domain,aspecialLipschitzdomain,andanunboundeddomainwithcom-
a
pact Lipschitzboundary. WeconstructtheNeumannGreen’sfunctionin
m
Qundertheassumptionthatweaksolutionsofthesystemssatisfyaninte-
[ riorH¨oldercontinuityestimate. WealsoestablishglobalGaussianbounds

forNeumannGreen’sfunctionunderanadditionalassumptionthatweak
1
solutions with zero Neumann data satisfy a local boundedness estimate.
v
7 In thescalar case, such a local boundedness estimate is a consequence of
3 De Giorgi-Moser-Nash theory holds for equations with bounded measur-
5 ablecoefficientsinSobolevextensiondomains,whileinthevectorialcase,
1 onemayneedtoimposefurtherregularityassumptionsonthecoefficients
. of the system as well as on the domain to obtain such an estimate. We
1
0 present a unified approach valid for both the scalar and vectorial cases
3 and discuss some applications of our results including the construction
1 of Neumann functions for second order elliptic systems with measurable
: coefficients in two dimensional domains.
v
i
X AMS subject classifications. 35K08,35K10,35K40
Key words. Green function; Neumannfunction; Neumann heat kernel; Neumannproblem;
r
a parabolicsystem;measurablecoefficients
1 Introduction
Inthisarticle,weareconcernedwithGreen’sfunctionforparabolicsystemswith
Neumann boundary condition, which hereafter we shall refer to as Neumann
Green’sfunction. ItisoftencalledNeumannheatkerneliftheparabolicsystem
˚ThisworkwassupportedNRFgrantsNo. 2010-0008224andR31-10049(WCUprogram).
:Department of Mathematics, Yonsei University, Seoul 120-749, Republic of Korea
([email protected]).
;DepartmentofComputationalScienceandEngineering,YonseiUniversity,Seoul120-749,
RepublicofKorea([email protected]).
1
has time-independent coefficients. In a cylindrical domain Ω , , where
the base Ω is an open connected set in Rn, we consider parˆabpo´li8c d8iffqerential
operator of the form
u u
Lu B B Aαβ x,t B . (1.1)
“ t ´ x p q x
B B α ˆ B β˙
Here, we use the summation convention over repeated indices 1 α,β n, u
is a column vector with components u1,...,uN, Aαβ x,t are Nď N mďatrices
p q ˆ
whoseelementsaαβ x,t areboundedmeasurablefunctionsdefinedontheentire
ij p q
space Rn`1 such that for arbitrary N n matrices ξ, η with elements ξi, ηi,
ˆ α α
we have
(cid:12) (cid:12)
λ|ξ|2 aαβξjξi, (cid:12)aαβξjηi(cid:12) λ´1|ξ||η|, (1.2)
ď ij β α (cid:12) ij β α(cid:12)ď
where λ is positive constant and |ξ|2 ξiξi. We emphasize that we allow
“ α α
coefficients to be non-symmetric and time-dependent. By a Neumann Green’s
functionforthe system(1.1)inΩ , ,wemeananN N matrixvalued
ˆp´8 8q ˆ
function N x,t,y,s that is a solution of the initial-boundary value problem
p q
L N x,t,y,s 0 in Ω s, ,
x,t
p q“ ˆp 8q
B N x,t,y,s 0 on Ω s, ,
$ Bνx p q“ B ˆp 8q
N x,t,y,s δ x I on Ω t s ,
& y
p q“ p q ˆt “ u
where ν is t%he natural co-normal derivative, δ is Dirac delta function
y
B{B p¨q
concentrated at y, I is the N N identity matrix, and the equation as well as
ˆ
theboundaryconditionshouldbeinterpretedinsomeweaksense. Equivalently,
it can be defined as a function that weakly satisfies
L N x,t,y,s δ x δ t I in Ω , ,
x,t y s
p q“ p q p q ˆp´8 8q
B N x,t,y,s 0 on Ω , ,
$ Bνx p q“ B ˆp´8 8q
N x,t,y,s 0 on Ω ,s .
&
p q” ˆp´8 q
More preci%se definition of Neumann Green’s function is givenin Section 2.4. In
the case when the coefficients are time-independent, K x,y,t – N x,t,y,0
p q p q
is usually called a Neumann heat kernel.
We prove that if the base Ω is such that the multiplicative embedding in-
equality (3.1) holds there and if weak solutions of the system (1.1) satisfy an
interiorHo¨ldercontinuityestimate(3.2),thentheNeumannGreen’sfunctionex-
istsandsatisfiesanaturalgrowthestimatenearthepole;seeTheorem3.9. The
multiplicative embedding inequality (3.1) holds if, for example, Ω is a Sobolev
extensiondomainwithfiniteLebesguemeasure,aspecialLipschitzdomain,and
an unbounded domain with compact Lipschitz boundary, etc. Also, the Ho¨lder
continuity estimate (3.2) holds, for example, in the scalar case (i.e., N 1), in
“
the case when n 2 and the coefficients are time-independent, and in the case
“
when the coefficients belong to the VMO class; see Section 4.
We are also interested in the following global Gaussian estimate for the
Neumann Green’s function: For any T 0, there exists a positive constant C
ą
2
such that for all t,s satisfying 0 t s T and x,y Ω, we have
ă ´ ă P
C κ|x y|2
|N x,t,y,s | exp ´ , (1.3)
p q ď t s n{2 ´ t s
p ´ q " ´ *
whereκisapositiveconstantindependentofT. Wepresenthowonecanderive
theglobalestimatelike(1.3)usingalocalboundednessestimate(3.18)forweak
solutions of the system with zero Neumann data; see Theorem 3.21. It is well
knownthatthelocalboundednessestimate(3.18)isavailable,forexample,when
N 1andΩisa boundedextensiondomainorwhenn 2,the coefficientsare
“ “
time-independent, and Ω is a bounded Lipschitz domain, etc.; see Section 4. In
fact, we show that the local boundedness estimate is a necessary and sufficient
condition for the system to have a global Gaussian estimate for the Neumann
Green’s function; see Theorem 3.24.
TheGreen’sfunctionforaparabolicequation(i.e.,N 1)withrealmeasur-
“
ablecoefficientsinthefreespacewasfirststudiedbyNash[24]anditstwo-sided
Gaussian bounds were later obtained by Aronson [2]. There is a vast literature
on Green’s function for parabolic equations satisfying Dirichlet or Neumann
conditions. In the case when the coefficients are time-independent, they are
called Dirichlet or Neumann heat kernel and have been studied by many au-
thors; see Davies [12] and references therein, [26, 31] for a treatment of heat
kernels on non-Euclidean setting, and also a very recent article [18]. It is well
known that Aronson’s bounds are no longer available for a parabolic equation
with complex valued coefficient when n 3. Related to Kato square root
ě
problem, Auscher [3] obtained an upper Gaussian bound of the heat kernel for
an elliptic operator whose coefficients are complex L8-perturbation of real co-
efficients; see [4, 5, 6] and also [1, 11, 25] for related results. The Dirichlet
Green’s function for a parabolic system with time-dependent, measurable co-
efficients was studied in papers by Cho et al. [8, 9], where Dirichlet Green’s
functionis constructedinanycylindricaldomainunder anassumptionthatthe
weak solutions of the system has an interior Ho¨lder continuity estimate that
is equivalent to ours. In many respects, this article can be considered as nat-
ural follow-up of their papers. However, we must point out it requires some
effort and caution to handle Neumann condition and there are certainly some
subtleties involved here. The novelty of our paper lies in the following points.
First, we allow time-dependent, non-symmetric coefficients and do not specifi-
callyassumeanyregularityonthem. Secondly,wetreatparabolicequationsand
systems simultaneously and do not rely on methods that are specific to single
equationscase,suchasthemaximumprincipleorHarnack’sinequality. Finally,
our method separates existence of Neumann Green’s function from its global
estimates, which enables us to construct Neumann Green’s functions in a large
class of domains, where global Gaussian estimates for Neumann functions may
not be available anymore. In this paper, we only deal with so called strongly
ellipticsystemsasitisdescribedin(1.2),butourmethodcanbeaswellapplied
to systems satisfying a weaker condition; see [29] for the Neumann heat kernel
for the elliptic system of linear elasticity. For the heat kernel for second-order
3
ellipticoperatorsindivergenceformsatisfyingRobin-typeboundaryconditions,
we bring attention to a very recent article [16].
The organizationofthe paper is as follows. InSection 2, we introduce some
notation and definitions including the precise definition of Neumann Green’s
function. In Section 3, we state our three main theorems, which we briefly
describedabove. InSection4,wegiveexamplesforwhichourmainresultsapply,
and also in Theorem 4.5, we construct the Neumann function for second order
elliptic systems with measurable coefficients in an extension domain Ω R2
Ă
and obtain logarithmic bound for it. We give the proofs for our main results
and Theorem 4.5 in Section 5.
2 Preliminaries
2.1 Basic notation and definitions
We use X x,t to denote a point in Rn`1; x x ,...,x will always be
1 n
a point in “Rnp. Wqe also write Y y,s , X “xp ,t , etc.q We define the
0 0 0
parabolic distance between the poi“ntspX q x,t“anpd Y q y,s in Rn`1 as
“p q “p q
|X Y|P –max |x y|, |t s| ,
´ p ´ ´ q
a
where | | denotes the usual Euclidean norm. We write |X|P |X 0|P. For
an open¨set Q Rn`1, we denote “ ´
Ă
dX dist X, pQ inf |X Y|P: Y pQ ; inf ,
“ p B q“ ´ PB H“8
where Q denotes the usual par abolic boundary of Q(. We use the following
p
notionsBfor basic cylinders in Rn`1:
Q´ X B x t r2,t ,
rp q“ rp qˆp ´ q
Q` X B x t,t r2 ,
rp q“ rp qˆp ` q
Q X B x t r2,t r2 ,
r r
p q“ p qˆp ´ ` q
where B x is the usual Euclidean ball of radius r centered at x Rn. We use
r
p q P
the notation
1
u u.
“ |Q|ˆ
Q Q
We denote a b min a,b and a b max a,b .
^ “ p q _ “ p q
2.2 Function spaces
For spaces of functions defined on a domain Ω Rn, we use the same notation
Ă
as used in Gilbarg and Trudinger [17], while for those defined on a domain
Q Rn`1, we borrow notation mainly from Ladyzhenskaya et al. [22]. To
avoĂidconfusion,spacesoffunctionsdefinedonQ Rn`1 willbealwayswritten
in script letters throughout the article. For q Ă1, we let L Q denote the
q
ě p q
4
classical Banach space consisting of measurable functions on Q that are qth-
integrable. L Q is the Banach space consisting of all measurable functions
q,r
p q
on Q with a finite norm
b r{q 1{r
kukLq,rpQq “˜ˆa ˆˆΩ|upx,tq|qdx˙ dt¸ ,
where q 1 and r 1. L Q will be denoted by L Q . By Cµ,µ{2 Q we
q,q q
ě ě p q p q p q
denote the set of all bounded measurable functions u on Q for which |u|
µ,µ{2;Q
is finite, where we define the parabolic Ho¨lder norm as follows:
|u| u |u|
µ,µ{2;Q “r sµ,µ{2;Q` 0;Q
|u X u Y |
– sup p q´ p q sup|u X |, µ 0,1 .
|X Y|µ ` p q Pp s
X,YPQ P XPQ
´
X‰Y
Wewriteu C8 Q (resp. C8 Q¯ )ifuisaninfinitelydifferentiablefunctionon
Rn`1 with aPcocmppaqct supporct ipn Qq (resp. Q¯). We write D u D u u x
i “ xi “B {B i
(i 1,...,n) and u u t. We also write Du D u for the column vector
t x
“ “B {B “
D u,...,D u T. We write Q t for the set of allpoints x,t in Q and I Q
1 n 0 0
p q p q p q p q
for the set of all t such that Q t is nonempty. We denote
p q
u 2 |D u|2dxdt esssup |u x,t |2dx.
~ ~Q “ˆ x ` ˆ p q
Q tPIpQq Qptq
ThespaceW1,0 Q denotestheBanachspaceconsistingoffunctionsu L Q
with weak deqrivpatiqves D u L Q (i 1,...,n) with the norm P qp q
i q
P p q “
kukWq1,0pQq “kukLqpQq`kDxukLqpQq
and by W1,1 Q the Banach space with the norm
q p q
kukWq1,1pQq “kukLqpQq`kDxukLqpQq`kutkLqpQq.
InthecasewhenQhasafiniteheight(i.e., Q Rn T,T forsomeT ),
we define V Q as the Banach space consisĂting ˆofp´all eleqments of W1ă,08Q
having a fini2tpe nqorm kukV2pQq – ~u~Q and the space V21,0pQq is obtain2edpbyq
completing the set W1,1 Q in the norm of V Q . When Q has an infinite
height, we say that u2 Vp Qq (resp. V1,0 Q ) 2ifpuq V Q (resp. V1,0 Q )
P 2p q 2 p q P 2p Tq 2 p Tq
for all T 0, where Q Q |t| T , and u . Note that this
T Q
ą “ Xt ă u ~ ~ ă 8
definition allows that 1 V1,0 Ω , when |Ω| . Finally, we write
u L Q if u L QP1 2forpallˆQp1´8Q8aqnqd similarlyăde8fine W1,0 Q , etc.
P q,locp q P qp q Ť q,locp q
2.3 Weak solutions
For f L Q N and gα L Q N (α 1,...,n), we say that u is a weak
2,1 2
solutioPn of Lpuq f D gαPin Qp ifqu V“Q N and satisfies the identity
α 2
“ ´ P p q
u φ B u,φ f φ gα D φ (2.1)
´ˆ ¨ t`ˆ p q“ˆ ¨ `ˆ ¨ α
Q Q Q Q
5
for all φ C8 Q N, where we set
P c p q
B u,v aαβD ujD vi.
p q“ ij β α
The adjoint operator L˚ of L is given by
u u
L˚u B B Aβα x,t T B . (2.2)
“´ t ´ x p q x
B B α ˆ B β˙
Note that the coefficients of (2.2) also satisfy the ellipticity and boundedness
condition (1.2) with the same constant λ. We say that u is a weak solution of
L˚u f D gα in Q if u V Q N and satisfies, for all φ C8 Q N, the
“ ´ α P 2p q P c p q
identity
u φ B φ,u f φ gα D φ. (2.3)
ˆ ¨ t`ˆ p q“ˆ ¨ `ˆ ¨ α
Q Q Q Q
Let Ω be an open set in Rn and Σ be a relatively open subset of Ω. Below we
B
denote Q Ω a,b and S Σ a,b , where a b . We say that
“ ˆp q “ ˆp q ´8ď ă ď8
u is a weak solution of
Lu f D gα in Q, u ν gαν on S
α α
“ ´ B {B “
ifu V Q N andsatisfiestheidentity (2.1)forallφ C8 Q S N. Similarly,
P 2p q P c p Y q
we say that u is a weak solution of
L˚u f D gα in Q, u ν˚ gαν on S
α α
“ ´ B {B “
if u V Q N and satisfies the identity (2.3) for all φ C8 Q S N. For
ψ LP2 Ω2pNqand a , we say that u is a weak solutioPn incVp1,0YQ Nq of the
P p q ‰´8 2 p q
problem
Lu f D gα in Q
α
“ ´
u ν gαν on Q
$ α p
B {B “ B
’& u ,a ψ on Ω
p¨ q“
if uPV21,0pQqN and sati’%sfies the identity
u φ B u,φ f φ gα D φ ψ φ ,a
´ˆ ¨ t`ˆ p q´ˆ ¨ ´ˆ ¨ α “ˆ ¨ p¨ q
Q Q Q Q Ω
for all φ C8 Q¯ N that is equal to zero for t b. For b , we say that u is
a weak soPlutcionp inq V1,0 Q N of the problem “ ‰8
2 p q
L˚u f D gα in Q
α
“ ´
u ν˚ gαν on Q
$ α p
B {B “ B
’& u ,b ψ on Ω.
p¨ q“
if u V1,0 Q N and sat’%isfies the identity
P 2 p q
u φ B φ,u f φ gα D φ ψ φ ,b
ˆ ¨ t`ˆ p q´ˆ ¨ ´ˆ ¨ α “ˆ ¨ p¨ q
Q Q Q Q Ω
for all φ C8 Q¯ N that is equal to zero for t a.
P c p q “
6
2.4 Neumann Green’s function
Let Q – Ω , , where Ω is a domain in Rn. We say that an N N
ˆp´8 8q ˆ
matrix valued function N X,Y N x,t,y,s , with measurable entries
p q“ p q
N : Q Q , ,
ij
ˆ Ñr´8 8s
is Neumann Green’s function of L in Q if it satisfies the following properties:
a) ForallY Q,wehaveN ,Y W1,0 Q N2 andN ,Y V Q Q Y N2
P p¨ qP 1,locp q p¨ qP 2p z rp qq
for any r 0.
ą
b) For all Y Q, we have LN ,Y δ I in Q, B N ,Y 0 on Q, in
the sense tPhat for any φ C8p¨Q¯ qN“, weYhave Bν p¨ q “ Bp
P c p q
N ,Y φi aαβD N ,Y D φi φk Y . (2.4)
ˆ ´ ikp¨ q t` ij β jkp¨ q α “ p q
Q
´ ¯
c) For any f f1,...,fN T C8 Q¯ N, the function u given by
“p q P c p q
u X – N Y,X Tf Y dY
p q ˆ p q p q
Q
is a weak solution in V1,0 Q N of L˚u f in Q, u ν˚ 0 on Q.
2 p q “ B {B “ Bp
We note thatpartc)ofthe abovedefinitiongivesthe uniqueness ofa Neumann
Green’s function.
3 Main results
WemakethefollowingassumptionstoconstructtheNeumannGreen’sfunction
in Q–Ω , .
ˆp´8 8q
A1. We assume that C8 Ω¯ is dense in W1,2 Ω and there exists a constant γ
such that for all uc pW˜q1,2 Ω , we have p q
P p q
kukL2pn`2q{npΩq ďγkDukLn{2ppnΩ`q2qkukL2{2ppnΩ`q2q, (3.1)
where we use the notation
W1,2 Ω if |Ω|
W˜1,2 Ω – p q “8
p q # u W1,2 Ω : u 0 if |Ω| .
P p q Ω “ ă8
´
(
A2. There exist µ 0,1 , R 0, , and B 0 such that if u is a weak
0 c 0
solution of LuP p0 (rsesp. LP ˚pu8s0) in Q ąQ´ X (resp. Q Q` X ),
where X Q an“d 0 R d¯ –“d R ,“thenRup isqHo¨lder con“tinuRopusqin
X X c
1Q Q´P X (respă. 1Qă Q` X^) with an estimate
2 “ R{2p q 2 “ R{2p q
1{2
rusµ0,µ0{2;21Q ďB0R´µ0 |upYq|2dY . (3.2)
ˆ Q ˙
7
Remark 3.3. In the case when n 3, the inequality
ě
kukL2n{pn´2qpΩq ďCkDukL2pΩq (3.4)
implies (via Ho¨lder’s inequality) the inequality (3.1) and also the inequality
kukL2pΩq ďCkDukLn{2ppnΩ`q2qkukL2{1ppnΩ`q2q. (3.5)
IfΩ Rn,then(3.4)and(3.5)aretheSobolevinequalityandtheNashinequal-
“
ity respectively. In fact, they are equivalent in that case; see the monograph
by Saloff-Coste [28]. From this point of view, one may think A1 roughly means
that Ω behaves like Rn with respect to W1,2 functions.
Remark 3.6. The inequality (3.1) implies the following: If u V Ω a,b is
2
such that u ,t W˜1,2 Ω for a.e. t a,b , where a P b p ˆ,pthenqqwe
p¨ q P p q P p q ´8 ď ă ď 8
have (see [22, pp. 74–75])
kukL2pn`2q{npΩˆpa,bqq ďγ~u~Ωˆpa,bq. (3.7)
Another important consequence of (3.1) is that for u V Ω a,b with
2
P p ˆ p qq
b a , we have
´ ă8
1
kukL2pn`2q{npΩˆpa,bqq ď 2γ`pb´aqn{2|Ω|´1 n`2 ~u~Ωˆpa,bq. (3.8)
´ ¯
We refer to [22, Eq. (3.8), p. 77] for the proof of (3.8).
Theorem 3.9. Assume the conditions A1 and A2. Then there exists a unique
Neumann Green’s function N X,Y N x,t,y,s of L in Q–Ω , .
p q“ p q ˆp´8 8q
It is continuous in X,Y Q Q: X Y and satisfies N x,t,y,s 0 for
t s. For any f tpf1,..q.,PfNˆT C8‰Q¯ Nu, the function u pgiven byq“
ă “p q P c p q
u X – N X,Y f Y dY (3.10)
p q ˆ p q p q
Q
is a weak solution in V1,0 Q N of
2 p q
Lu f in Q, u ν 0 on Q. (3.11)
p
“ B {B “ B
Moreover, for all ψ ψ1,...,ψN T L2 Ω N, the function given by
“p q P p q
u x,t N x,t,y,s ψ y dy (3.12)
p q“ˆ p q p q
Ω
is a unique weak solution in V1,0 Ω s, N of the problem
2 p ˆp 8qq
Lu 0 in Ω s,
“ ˆp 8q
u ν 0 on Ω s, (3.13)
$
B {B “ B ˆp 8q
’&u ,s ψ on Ω.
p¨ q“
’%
8
Furthermore, for X,Y PQ satisfying 0ă|X ´Y|P ă 12d¯Y, we have
|N X,Y | C|X Y|´n, (3.14)
P
p q ď ´
and for X,X1,Y PQ satisfying 2|X´X1|P ă|X ´Y|P ă 21d¯Y,
|N X,Y N X1,Y | C|X X1|µ0|X Y|´n´µ0, (3.15)
P P
p q´ p q ď ´ ´
where C C n,N,λ,µ ,B .
0 0
“ p q
Remark 3.16. In the proof of Theorem 3.9 we will see that for any Y Q and
0 R d¯ , we have P
Y
ă ă
i) ~Np¨,Yq~QzQ¯RpYq ďCR´n{2.
ii) kN˜p¨,YqkL2pn`2q{npQzQ¯RpYqq ďCγR´n{2,
iii) (cid:12)(cid:12) X PQ: |N˜pX,Yq|ąτ (cid:12)(cid:12)ďCγτ´nn`2, @τ ąd¯´Yn.
iv) (cid:12)(cid:12) X PQ: |DxNpX,Yq|ą(τ (cid:12)(cid:12)ďCτ´nn``12, @τ ąd¯´Ypn`1q.
v) k Np¨,YqkLppQRpYqq ďCp,γR(´n`pn`2q{p, @pPr1,n`n2q.
vi) kDxNp¨,YqkLppQRpYqq ďCpR´n´1`pn`2q{p, @pPr1,nn``12q.
Here, we denote N˜ ,Y N ,Y 1 H t s I, where H t 1 t and
p¨ q“ p¨ q´ |Ω| p ´ q p q“ r0,8qp q
we use the convention 1 0.
{8“
Remark 3.17. If Ω Rn is such that there exists a bounded trace operator
fromW1,2 Ω to L2ĂΩ , then it canbe shownthat for f C8 Ω¯ a,b N and
g C8 Ωp q a,b Np ,qwhere a b , u definedPbyc p ˆr sq
P c pB ˆr sq ´8ă ă ă8
b b
u X N X,Y f Y dY N X,Y g Y dS
p q“ˆ ˆ p q p q `ˆ ˆ p q p q Y
a Ω a BΩ
is a unique weak solution in V1,0 Ω a,b N of the problem
2 p ˆp qq
Lu f in Ω a,b
“ ˆp q
u ν g on Ω a,b
$
B {B “ B ˆp q
’&u ,a 0 on Ω.
p¨ q“
Thefollowing“localb’%oundedness”assumptionisusedtoobtainglobalGaus-
sian estimates for the Neumann Green’s function.
A3. There exist R 0, , B 0 such that if u is a weak solution of
M 1
Pp 8s ą
Lu 0 in Q Q´ X Q, u ν 0 on S Q´ X Q,
“ “ Rp qX B {B “ “ Rp qXBp
resp. L˚u 0 in Q Q` X Q, u ν˚ 0 on S Q` X Q,
p “ “ Rp qX B {B “ “ Rp qXBp q
where X Q and 0 R R , then u is bounded in 1Q Q´ X Q
P ă ă M 2 “ R{2p qX
(resp. 1Q Q` X Q) and we have an estimate
2 “ R{2p qX
kukL8p12Qq ďB1R´pn`2q{2kukL2pQq. (3.18)
9
Remark 3.19. Note that A3 implies that there exists θ 0 such that for all
ą
x Ω and 0 R R , we have
M
P ă ă
θRn |Ω B x |, (3.20)
R
ď X p q
which follows from (3.18) by taking u 1 and θ 1 B2. In particular, we
“ “ { 1
should have R in the case when |Ω| . In fact, we may take R
M M
|Ω|1{n in that caseăw8ithout loss of generality.ă 8 “
Theorem 3.21. Assume that the conditions A1-A3. Then, for t s, we have
ą
the Gaussian bound for the Neumann Green’s function
C κ|x y|2
|N x,t,y,s | exp ´ ´ , (3.22)
p q ď t s R2 n{2 t s
tp ´ q^ Mu ˆ ´ ˙
where C C n,N,λ,B and κ κ λ 0. In particular, the estimate (3.22)
1
“ p q “ p qą
implies that for 0 t s T, we have
ă ´ ď
C κ|x y|2
|N x,t,y,s | exp ´ ´ , (3.23)
p q ď t s n{2 t s
p ´ q ˆ ´ ˙
where C C n,N,λ,B ,T .
1
“ p q
Finally, the next theorem says that the converse of Theorem 3.21 is also
true, and thus that the condition A3 is equivalent to a global bound (3.22) for
the Neumann Green’s function.
Theorem 3.24. Assume that the conditions A1 and A2. Suppose there exist
constants B and κ 0 such that the Neumann Green’s function has the bound
2
ą
B κ|x y|2
2
|N x,t,y,s | exp ´ ´ , (3.25)
p q ď t s R2 n{2 t s
tp ´ q^ Mu ˆ ´ ˙
for all t s and x,y Ω. Then the condition A3 is satisfied with the same R
M
ă P
and B depending on n,N,λ,κ, and B .
1 2
4 Applications
4.1 Examples
4.1.1 Examples for A1
First, we give examples of domains satisfying the condition A1.
(1) We say that Ω is an extension domain (for W1,2 functions) if there exists a
linear operator E: W1,2 Ω W1,2 Rn such that
p qÑ p q
kEukL2pRnq ďCkukL2pΩq, kEukW1,2pRnq ďCkukW1,2pΩq. (4.1)
10