Scale-invariant boundary Harnack principle on inner uniform domains in fractal-type spaces

0.25 MB·English

by  Janna Lierl

Scale-invariant boundary Harnack principle on
inner uniform domains in fractal-type spaces
2
1
0
2 Janna Lierl

n
July 27, 2012
a
J

1
1 Abstract

] We prove a scale-invariant boundary Harnack principle (BHP) on in-
R
ner uniform domains in metric measure spaces. We prove our result in
P the context of strictly local, regular, symmetric Dirichlet spaces, without
. assumingthattheDirichletforminducesametricthatgeneratestheorig-
h
t inal topology on the metric space. Thus, we allow the underlying space
a to be fractal, e.g. theSierpinski gasket.
m
2010 Mathematics Subject Classification: 31C25, 60J60,60J45.
[
Keywords: Inner Uniform domains, Dirichlet space, metric measure space,

1 boundary Harnack principle.
v
Acknowledgement: ResearchpartiallysupportedbyNSFgrantDMS1004771.
6
Address: MalottHall,DepartmentofMathematics,CornellUniversity,Ithaca,
3
2 NY 14853,United States.
2
.
1
Introduction
0
2
1 Given the notion of harmonic functions, the boundary Harnack principle is a
: propertyofadomainthatprovidescontroloverthe ratiooftwoharmonicfunc-
v
tions in that domain near some part of the boundary where the two functions
i
X satisfy the Dirichlet boundary condition. Whether a given domain satisfies the
r boundary Harnack principle depends on the geometry of its boundary. We are
a
interested in the case when the domain is uniform or, more generally, inner
uniform.
AikawaandAnconaprovedascale-invariantboundaryHarnackprincipleon
uniform and inner uniform domains, respectively, in Euclidean space.
In [8], Gyrya and Saloff-Coste generalized Aikawa’s approach to uniform
domains in strictly local, regular, symmetric Dirichlet spaces of Harnack-type
that admit a carr´e du champ. Moreover, they deduced that the boundary
Harnackprinciplealsoholdsoninneruniformdomains,byconsideringtheinner
uniform domain as a uniform domain in a different metric space, namely the
completion of the inner uniform domain with respect to its inner metric.
1
In[10],weprovedtheboundaryHarnackprincipledirectlyoninneruniform
domains, when the underlying space is a strictly local, regular (possibly non-
symmetric) Dirichlet space of Harnack-type.
In this paper, we assume that the underlying space (X,d,µ,E,D(E)) is a
metric measure Dirichlet space in the sense of [5]. The parabolic Harnack in-
equalityPHI(Ψ)whichweassumetoholdonthespace,hasaspace-timescaling
that is captured by the function
sβ¯, if s≤1,
Ψ(s)=Ψβ,β¯(s)=(sβ, if s>1,
for some β,β¯ ≥ 2. In the classical case, Ψ(r) = r2. Different values of β and
β¯ allow for fractal-type spaces, such as the Sierpinski gasket. For example,
removing the bottom line in the Sierpinski gasket produces a subset that is an
inner uniform domain in the Sierpinski gasket.
In fractal-type spaces, the energy measure of the Dirichlet form can be sin-
gular with respect to the measure of the underlying metric measure space. We
thusdonotmakeanyassumptionsonthepseudo-metricinducedbythe Dirich-
let form, in fact we do not consider it at all. This is contrary to [8] and [10].
The main resultof this paper is the following scale-invariantboundary Har-
nack principle.
Theorem 0.1 Let Ω be an inner uniform domain in the metric measure space
X. Suppose that X is equipped with astrictly local, regular, symmetric Dirichlet
form that satisfies the parabolice Harnack inequality PHI(Ψ). Then there exist
constantsA ,A ∈(1,∞)suchthatfor anyξ ∈∂Ω,R=R(ξ,Ω)>0small, and
0 1
any two non-negative weak solutions u and v of the Laplace equation Lu=0 in
B (ξ,A r) with weak Dirichlet boundary condition along ∂Ω, we have
Ω 0
u(x) v(x)
≤A ,
u(x′) 1v(x′)
for all x,x′ ∈ B (ξ,r). Here B (ξ,r) = {z ∈ Ω : d (ξ,z) < r} and d is the
Ω Ω Ω Ω
intrinsic distance of the domain. The constant A depends only on the Harnack
1
constant H , on β, β¯, and on the inner-uniformity constants c , C .
0 u u
In Section 1, we review some general properties of Dirichlet spaces and lo-
cal weak solutions. In Section 2 we describe the parabolic Harnack inequality
and equivalent conditions that we impose on the space. In Section 3 we prove
estimates for the Dirichlet heat kernel on balls. After recalling the definition
and some properties of inner uniform domains, we give estimates for Green’s
functions on balls intersected with an inner uniform domain. In Section 4 we
give a proof of the boundary Harnack principle.
2
1 Local weak solutions and capacity
1.1 Local weak solutions
Let (X,d,µ,E,D(E)) be a metric measure Dirichlet space (MMD space). That
is, X is a connected, locally compact, separable, complete metric space and
the metric d is geodesic. µ is a non-negative Borel measure on X that is finite
on compact sets and positive on non-empty open sets. (E,D(E)) is a symmet-
ric, strictly local, regular Dirichlet form on L2(X,µ). See [7]. The semigroup
(P ) associated with (E,D(E)) is assumed to be conservative. We denote by
t t≥0
(L,D(L)) the infinitesimal generator of (E,D(E)).
There exists a measure-valued quadratic form dΓ defined by
1
fdΓ(u,u)=E(uf,u)− E(f,u2), ∀f,u∈D(E)∩L∞(X),
2
Z
and extended to unbounded functions by setting Γ(u,u) = lim Γ(u ,u ),
n→∞ n n
whereu =max{min{u,n},−n}. Using polarization,weobtaina bilinearform
n
dΓ. In particular,
E(u,v)= dΓ(u,v), ∀u,v ∈D(E).
Z
For U ⊂X open, set
F (U)={f ∈L2 (U):∀ compact K ⊂U, ∃f♯ ∈D(E),f =f♯ a.e.},
loc loc K
where L2 (U) is the space of functions that are locally in L2(U).(cid:12)(cid:12) For f,g ∈
loc
F (U) we define Γ(f,g) locally by Γ(f,g) = Γ(f♯,g♯) , where K ⊂ U is
loc K K
compact and f♯,g♯ are functions in D(E) such that f = f♯, g = g♯ a.e. on K.
(cid:12) (cid:12)
Furthermore, set (cid:12) (cid:12)
F(U)={u∈F (U): |u|2dµ+ dΓ(u,u)<∞},
loc
ZU ZU
F (U)={u∈F(U): The essential support of u is compact in U}.
c
Definition 1.1 Foranopen setU ⊂X,theDirichlet-typeform onU is defined
as
ED(f,g)=E(f,g), f,g ∈D(ED),
U U
where the domain D(ED)=F0(U) is defined as the closure of the space C∞(U)
U 0
of all smooth functions with compact support in U. The closure is taken in the
norm EUD,1(f,f)12 =(EUD(f,f)+kfk2)12.
The Dirichlet form (ED,D(ED)) is associated with a semigroup PD(t), t > 0.
U U U
Usingthereasoningin[14][Section2.4],onecanshowthatunderthehypothesis
that the underlying space satisfies the parabolic Harnack inequality PHI(Ψ),
the semigroup has a continuous kernel pD(t,x,y). Moreover, the map y 7→
U
pD(t,x,y) is in F0(U).
U
3
Definition 1.2 Let V ⊂U be open. Set
F0 (U,V)={f ∈L2 (V,µ):∀ open A⊂V rel. compact in U with
loc loc
d (A,U \V)>0, ∃f♯ ∈F0(U):f♯ =fµ-a.e. on A},
U
where
d (A,U\V)=inf length(γ) γ :[0,1]→U continuous,γ(0)∈A,γ(1)∈U\V .
U
Definition 1.3 Le(cid:8)t V ⊂ U (cid:12)(cid:12)be open and f ∈ Fc(V)′, the dual space of Fc(V(cid:9))
(identify L2(X,µ) with its dual space using the scalar product). A function
u:V →R is a local weak solution of the Laplace equation −Lu=f in V, if
(i) u∈F (V),
loc
(ii) For any function φ∈F (V), E(u,φ)= fφdµ.
c
If in addition R
u∈F0 (U,V),
loc
then u is a local weak solution with Dirichlet boundary condition along ∂U.
For a time interval I and a Hilbert space H, let L2(I → H) be the Hilbert
space of those functions v :I →H such that
1/2
kvk = kv(t)k dt <∞.
L2(I→H) H
(cid:18)ZI (cid:19)
LetW1(I →H)⊂L2(I →H)betheHilbertspaceofthosefunctionsv :I →H
in L2(I → H) whose distributional time derivative v′ can be represented by
functions in L2(I →H), equipped with the norm
1/2
kvk = kv(t)k +kv′(t)k dt <∞.
W1(I→H) H H
(cid:18)ZI (cid:19)
Let
F(I ×X)=L2(I →D(E))∩W1(I →D(E)′), (1)
where D(E)′ denotes the dual space of D(E). Similarly, define F0(I ×U) by
replacing D(E) by F0(U) in (1). Let
F (I ×U)
loc
be the set of all functions u:I×U →R such that for any open interval J that
isrelativelycompactinI,andanyopensubsetArelativelycompactinU,there
exists a function u♯ ∈F(I ×U) such that u♯ =u a.e. in J ×A. Let
F (I×U)={u∈F (I×U):u(t,·) has compact support in U for a.e. t∈I}.
c loc
4
For an open subset V ⊂U, let Q=I ×V and let
F0 (U,Q)
loc
be the set of all functions u : Q → R such that for any open interval J that is
relatively compact in I, and any open set A⊂V relatively compact in U¯ with
d (A,U \V)>0, there exists a function u♯ ∈F0(I ×U) such that u♯ =u a.e.
U
in J ×A.
Definition 1.4 Let I be an open interval and V ⊂U open. Set Q=I ×V. A
function u:Q→R is a local weak solution of the heat equation (∂ −L)u=0
t
in Q, if
(i) u∈F (Q),
loc
(ii) For any open interval J relatively compact in I,
∀φ∈F (Q), ∂ φudµdt− E(φ(t,·),u(t,·))dt =0.
c t
ZJZV ZJ
If in addition
u∈F0 (U,Q),
loc
then u is a local weak solution with Dirichlet boundary condition along ∂U.
1.2 Capacity
Let U ⊂ X be open and relatively compact. The extended Dirichlet space
F (U) is defined (see [7]) as the family of all measurable, almost everywhere
e
finitefunctionsusuchthatthereexistsanapproximatingsequenceu ∈D(ED)
n U
that is ED-Cauchy and u=limu almost everywhere.
U n
For any set A⊂U define
L ={u∈F (U):u˜≥1 q.e. on A},
A,U e
where u˜ is a quasi-continuous modification of u. If L 6= ∅, there exists
A,U
a unique function e ∈ L such that E(e ,w) ≥ E(e ,e ) holds for all
A A,U A A A
w∈L . The 0-capacity of A in U is defined by
A,U
E(e ,e ), L 6=∅
A A A,U
Cap (A)=
U (+∞, LA,U =∅.
By [6][Proposition VI.4.3], e = G ν , where ν is a finite measure with
A U A A
supp(ν ) contained in the completion of A, and G is the Green’s function
A U
associated with ED (see [6][page 256]). Thus, if A is compact,
U
Cap (A)=E(e ,e )=E(G ν ,e )= e dν =ν (A).
U A A U A A A A A
Z
5
2 Parabolic Harnack inequality and equivalent
conditions
Let (X,d,µ,E,D(E)) be a MMD space.
Definition 2.1 (X,µ)satisfiesthe volumedoublingpropertyVDifthereexists
a constant D ∈(0,∞) such that for every x∈X, R>0,
0
V(x,2R)≤D V(x,R),
0
where V(x,R)=µ(B(x,R)) denotes the volume of the ball B(x,R).
Let β,β¯≥2 and set
sβ¯, if s≤1,
Ψ(s)=Ψβ,β¯(s)=(sβ, if s>1.
Definition 2.2 (E,D(E)) satisfies the cut-off Sobolev inequality CS(Ψ) on X,
if there exist constants θ ∈ (0,1] and c ,c > 0 such that the following holds.
1 2
For any ball B(x ,2R)⊂X, there exists a cut-off function ψ(=ψ ) with the
0 x0,R
properties:
(i) ψ(x)≥1 for x∈B(x ,R/2).
0
(ii) ψ(x)=0 for x∈X \B(x ,R).
0
(iii) |ψ(x)−ψ(y)|≤c (d(x,y)/R)θ for all x,y ∈X.
1
(iv) For any ball B =B(x,s) with x∈X, 0<s≤R, and any f ∈D(E),
s 2θ
f2dΓ(ψ,ψ)≤c dΓ(f,f)+Ψ(s)−1 f2dµ ,
2
R
ZB (cid:16) (cid:17) (cid:18)Z2B Z2B (cid:19)
where 2B =B(x,2s).
Definition 2.3 (E,D(E)) satisfies the Poincar´einequalityPI(Ψ) on X if there
exists a constant P such that for any ball B =B(x,R) with x∈X, R>0, and
0
any f ∈D(E),
(f −f )2dµ≤P Ψ(R) dΓ(f,f),
B 0
ZB ZB
Here f =µ(B)−1 fdµ.
B B
Definition 2.4 (ER,D(E)) satisfies the capacityestimate CAP(Ψ) if there exist
constants c,C ∈(0,∞) such that for any x∈X, R≥0,
Ψ(R) −1 Ψ(R)
c ≤ Cap (B(x,R)) ≤C .
V(x,R) B(x,2R) V(x,R)
(cid:16) (cid:17)
6
For any x∈X, R>0, define
I = 0,4Ψ(R))
B =B(x,R)
(cid:0)
Q=I ×B(x,2R)
Q = Ψ(R),2Ψ(R) ×B
−
Q+ =(cid:0)3Ψ(R),4Ψ(R(cid:1)) ×B.
Definition 2.5 Theoperator(L(cid:0),D(L))satisfie(cid:1)sthe parabolicHarnackinequal-
ity PHI(Ψ) if there exists a constant H ∈ (0,∞) such that for any x ∈ X,
0
R>0, any positive function u∈F (Q) with (∂ −L)u=0 weakly in Q has a
loc t
quasi-continuous modification u˜ so that
sup u˜(z)≤H inf u˜(z).
0
z∈Q− z∈Q+
For (t,r)∈(0,∞)×[0,∞) we consider the two regions
Λ ={(t,r):t≤1∨r} and Λ ={(t,r):t≥1∨r}.
1 2
Let
rβ 1/(β−1)
h (r,t)=exp − .
β
t
(cid:18) (cid:19) !
Definition 2.6 (E,D(E)) satisfies HK(Ψ) if the heat kernel p(t,x,y) on X
exists and if there are positive constants c , c , c , c so that
1 2 3 4
c1hβ¯(c2d(x,y),t) ≤p(t,x,y)≤ c3hβ¯(c4d(x,y),t),
V(x,t1/β¯) V(x,t1/β¯)
for x,y ∈X and t∈(0,∞) with (t,d(x,y))∈Λ , and
1
c h (c d(x,y),t) c h (c d(x,y),t)
1 β 2 3 β 4
≤p(t,x,y)≤ ,
V(x,t1/β) V(x,t1/β)
for x,y ∈X and t∈(0,∞) with (t,d(x,y))∈Λ .
2
Theorem 2.7 Let (X,µ,E,D(E)) be a MMD space. The following are equiva-
lent:
(i) (E,D(E)) satisfies PHI(Ψ).
(ii) (E,D(E)) satisfies HK(Ψ).
(iii) (E,D(E)) satisfies VD, PI(Ψ) and CS(Ψ).
Proof. The equivalence of (i) and (ii) is proved in [9][Theorem 5.3] for suffi-
ciently regularsolutions, andin [4]. The equivalence of (i) and (iii) is provedin
[5][Theorem 2.16]. (cid:3)
Theorem 2.8 Let (X,µ,E,D(E)) be a MMD space. Suppose that (E,D(E))
satisfies PHI(Ψ). Then (E,D(E)) satisfies CAP(Ψ).
Proof. See [5]. (cid:3)
7
3 Inner uniform domains and Green’s function
estimates
Let (X,d,µ,E,D(E)) be a MMD space that satisfies PHI(Ψ).
3.1 (Inner) uniformity
Let Ω⊂X be open and connected. The inner metric on Ω is defined as
d (x,y)=inf length(γ) γ :[0,1]→Ω continuous,γ(0)=x,γ(1)=y .
Ω
Let Ω˜ be the com(cid:8)pletion of(cid:12)(cid:12)Ω with respect to dΩ. Whenever we consid(cid:9)er an
inner ball B (x,R) = {y ∈ Ω˜ : d (x,y) < R}, we assume that its radius is
Ω˜ Ω
minimal in the sense that B (x,R)6= B (x,r) for all r <R. If x is a point in
Ω˜ Ω˜
Ω, denote by δ(x) = δ (x) = d(x,∂Ω) the distance from x to the boundary of
Ω
Ω.
Definition 3.1 (i) Let γ : [α,β] → Ω be a rectifiable curve in Ω and let
c∈(0,1), C ∈(1,∞). We call γ a (c,C)-uniform curve in Ω if
δ γ(t) ≥c·min d γ(α),γ(t) ,d γ(t),γ(β) , for all t∈[α,β], (2)
Ω
(cid:0) (cid:1) (cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9)
and if
length(γ)≤C·d γ(α),γ(β) .
The domain Ω is called (c,C)-uniformi(cid:0)f any twop(cid:1)oints in Ω can be joined
by a (c,C)-uniform curve in Ω.
(ii) Inner uniformity is defined analogously by replacing the metric d on X
with the inner metric d on Ω.
Ω
(iii) The notion of (inner) (c,C)-length-uniformity is defined analogously by
replacing d(γ(a),γ(b)) by length(γ ).
[a,b]
(cid:12)
Thefollowingpropositionistakenfrom[8(cid:12)][Proposition3.3]. Seealso[12][Lemma
2.7].
Proposition 3.2 Assumethat(X,d)isacomplete, locally compact lengthmet-
ricspacewiththepropertythatthereexistsaconstantD suchthatforanyr>0,
themaximalnumberofdisjoint ballsofradiusr/4containedinanyballofradius
r is bounded above by D. Then any connected open subset U ⊂X is uniform if
and only if it is length-uniform.
Lemma 3.3 Let Ω be a (c ,C )-inner uniform domain in (X,d). For every
u u
ball B = B (x,r) in (Ω˜,d ) with minimal radius, there exists a point x ∈ B
Ω˜ Ω r
with d (x,x )=r/4 and d(x ,Ω˜ \Ω)≥c r/8.
Ω r r u
8
Proof. See [8][Lemma 3.20]. (cid:3)
The following lemma is crucial for the proof of the boundary Harnack prin-
ciple on inner uniform domains rather than uniform domains. A versionof this
lemma was already used in [2] to prove a boundary Harnack principle on inner
uniform domains in Euclidean space.
For any ball D =B(x,r) with x∈Ω˜, let D′ be the connected component of
D∩Ω˜ that contains x.
Lemma 3.4 LetΩ bea(c ,C )-inner uniform domain in (X,d,µ)andassume
u u
that µ has the volume doubling property with constant D . Then there exists a
0
positive constant C such that for any ball D =B(x,r) with x∈Ω˜, r >0,
0
B (x,r)⊂D′ ⊂B (x,C r).
Ω˜ Ω˜ 0
The constant C depends only on D , c , C .
0 0 u u
Proof. See [10][Lemma 3.7]. (cid:3)
3.2 Green’s function estimates
Theorem 3.5 (i) For any fixed ǫ∈(0,1) there are constants c ,C ∈(0,∞)
1 1
such that for any ball B =B(a,R)⊂X and any x,y ∈B(a,(1−ǫ)R) and
0<ǫt≤Ψ(R), the Dirichlet heat kernel pD, is bounded below by
B
C
pDB(t,x,y)≥ V(x,t1/β1¯∧R )hβ¯(c1d(x,y),t) t≤1∨d(x,y), (3)
x
C
pD(t,x,y)≥ 1 h (c d(x,y),t) t≥1∨d(x,y), (4)
B V(x,t1/β ∧R ) β 1
x
where R =d(x,∂B).
x
(ii) For any fixed ǫ ∈ (0,1) there are constants c ,C ∈ (0,∞) such that for
2 2
any ball B = B(a,R) ⊂ X and any x,y ∈ B(a,R), t ≥ ǫΨ(R), the
Dirichlet heat kernel pD is bounded above by
B
C
pD(t,x,y)≤ 2 exp(−c t/Ψ(R)). (5)
B V(x,R) 2
(iii) There existconstantsc ,C ∈(0,∞)suchthatfor anyballB =B(a,R)⊂
3 3
X and any x,y ∈ B(a,R), the Dirichlet heat kernel pD is bounded above
B
by
pD(t,x,y)≤ C3hβ¯(c3d(x,y),t), t≤1∨d(x,y) (6)
B V(x,t1/β¯)
C h (c d(x,y),t)
pD(t,x,y)≤ 3 β 3 , t≥1∨d(x,y). (7)
B V(x,t1/β)
All the constants c ,C above depend only on H , β, β¯.
i i 0
9
Proof of Theorem 3.5. Fix a ball B =B(a,R)⊂X. (iii) Let x,y ∈B(a,R).
By Theorem 2.7, we have for the heat kernel p(t,x,y) on X associated with
(E,D(E)),
p(t,x,y)≤ c3hβ¯(c4d(x,y),t), t≤1∨d(x,y) (8)
V(x,t1/β¯)
c h (c d(x,y),t)
3 β 4
p(t,x,y)≤ , t≥1∨d(x,y). (9)
V(x,t1/β)
Thus, assertion (iii) follows from the set monotonicity of the kernel pD.
B
To show the on-diagonal estimate in (i) we follow [13][Proof of Theorem
5.4.10]. Letx∈B(a,(1−ǫ)R)and0<ǫt≤Ψ(R). Lett′ = t ∧Ψ(R /8). Then
4 x
B = B(x,8Ψ−1(t′)) lies in B(a,R). By the cut-off Sobolev inequality which
x
holds by Theorem 2.7, there exists a cut-off function ψ such that 0 ≤ ψ ≤ 1,
ψ =1 on B(x,2Ψ−1(t′)) and ψ =0 on X \B(x,4Ψ−1(t′)). Let
PD ψ(y) if t>0,
u(t,y)= Bx,t
(ψ(y) if t≤0.
Then u is a local weak solution of
∂
Lu= u on Q′ =(−∞,+∞)×B(x,2Ψ−1(t′)).
∂t
Applying the parabolic Harnack inequality to u and then to pD (·,x,·), we
Bx
get
1=u(0,x)≤Cu(t/2,x)
=C pD (t/2,x,y)ψ(y)µ(dy)
Bx
Z
≤C pD (t/2,x,y)µ(dy)
ZB(x,4Ψ−1(t′)) Bx
≤C′V(x,4Ψ−1(t′))pD (t,x,x).
Bx
Using the setmonotonicity ofthe kernelandthe doubling propertyofthe func-
tion Ψ and the volume, we get
1
pD(t,x,x)≥pD (t,x,x)≥C′′ .
B Bx V(x,Ψ−1(t)∧R )
x
The off-diagonal estimate now follows from [9][Lemma 5.1].
(ii) follows from changing notation in [9][Lemma 5.13 part 3]. (cid:3)
Lemma 3.6 For any relatively compact open set V ⊂ X, the Green function
y 7→G (x,y) is in F0 (V,V \{x}) for any fixed x∈V.
V loc
10