Wiener-Landis criterion for Kolmogorov-type operators

0.24 MB·

by  A. E. Kogoj

WIENER-LANDIS CRITERION
FOR KOLMOGOROV-TYPE OPERATORS
7
1 ALESSIA KOGOJ,ERMANNOLANCONELLI,ANDGIULIOTRALLI
0
2
Abstract. Weestablishanecessaryandsufficientconditionforaboundarypointtobe

n regular for the Dirichletproblem related to a class of Kolmogorov-type equations. Our
a criterionisinspiredbytwoclassicalcriteriaforthe heatequation: the Evans–Gariepy’s
J Wienertest,andacriterionbyLandisexpressedintermsofaseriesofcaloricpotentials.

4

]
P
A 1. Introduction
. Aim of this paper is to establish a necessary and sufficient condition for the regularity
h
t of a boundary point for the Dirichlet problem related to a class of hypoelliptic evolution
a
equationsofKolmogorov-type. OurcriterionisinspiredbothtotheEvans–Gariepy’sWiener
m
testforthe heatequation,andto acriterionbyLandis,forthe heatequationtoo,expressed
[
in terms of a series of caloric potentials.

1 The partial differential operators we are dealing with are of the following type
v
(1.1) =div(A )+ Bx, ∂ ,
3 L ∇ h ∇i− t
7 where A = (a ) and B = (b ) are N N real and constant matrices,
i,j i,j=1,...,N i,j i,j=1,...,N
0 z =(x,t)=(x ,...,x ,t) is the point of RN+1, =(∂ ,×...,∂ ), div and , stand for
01 the gradient, t1he diverNgence and the inner produ∇ct in RxN1, respexctNively. h i
. The matrix A is supposed to be symmetric and positive semidefinite. Moreover,letting
1
0 E(s):=exp( sB), s R,
7 − ∈
1 we assume that the following Kalman condition is satisfied: the matrix
v: t
C(t)= E(s)AET(s)ds,
i
X Z0
r is strictly positive definite for every t>0. As it is quite well known, the conditionC(t)>0
a
fort>0isequivalenttothehypoellipticityof in(1.1),i.etothesmoothnessofuwhenever
L
u is smooth (see, e.g., [9]). We also assume the operator to be homogeneous of degree
Ltwo with respect to a group of dilations in RN+1. As weLwill recall in Section 2, this is
equivalent to assume A and B taking the blocks form (2.1) and (2.2).
Undertheaboveassumptions,onecanapplyresultsandtechniquesfrompotentialtheory
in abstractHarmonic Spaces, as presented, e.g, in [2]. As a consequence,for every bounded
open set Ω RN+1 and for every function f C(∂Ω,R), the Dirichlet problem
⊆ ∈
(1.2) u=0 in Ω, u =f,
∂Ω
L |
2010 Mathematics Subject Classification. 35H10, 31C15,35K65.
Key words and phrases. Kolmogorovoperators,Potential analysis,Wienertest.
1
2 A.E.KOGOJ,E.LANCONELLI,ANDG.TRALLI
has a generalized solution HΩ in the sense of Perron–Wiener–Brelot–Bauer. The function
f
HΩ is smooth and solves the equationin (1.2) in the classicalsense. However,it may occur
f
that HΩ does not assume the boundary datum. A point z ∂Ω is called -regular for Ω if
f 0 ∈ L
lim HΩ(z)=f(z ) f C(∂Ω,R).
z→z0 f 0 ∀ ∈
Aimofthispaperistoobtainacharacterizationofthe -regularboundarypointsinterms
L
ofaserieinvolving -potentialsofregionsinΩc,the complementofΩ,within differentlevel
L
sets of Γ, the fundamental solution of . More precisely, if z ∂Ω and λ ]0,1[ are fixed,
0
we define for k N L ∈ ∈
∈
1 klogk 1 (k+1)log(k+1)
Ωc(z )= z Ωc : Γ(z ,z) z .
k 0 ( ∈ (cid:18)λ(cid:19) ≤ 0 ≤(cid:18)λ(cid:19) )∪{ 0}
Then, our main result is the following
Theorem 1.1. Let Ω be a bounded open subset of RN+1 and let z ∂Ω. Then z is
0 0
∈
-regular for ∂Ω if and only if
L
∞
(1.3) V (z )=+ .
Ωck(z0) 0 ∞
k=1
X
Here and in what follows, if F is a compact subset of RN+1, V will denote the -
F
L
equilibrium potential of F, and cap(F) will denote its -capacity. We refer to Section 3 for
L
the precise definitions.
From Theorem 1.1, one easily obtains a corollary resembling the Wiener test for the
classical Laplace and Heat operators.
Corollary 1.2. Let Ω be a bounded open subset of RN+1 and z ∂Ω. The following
0
∈
statements hold:
(i) if
∞ cap(Ωck(z0)) =+
λklogk ∞
k=1
X
then z is -regular;
0
L
(ii) if z is -regular then
0
L
∞ cap(Ωck(z0)) =+ .
λ(k+1)log(k+1) ∞
k=1
X
We can make the sufficient condition for the -regularity more concrete and more geo-
L
metrical with the following corollary.
Corollary 1.3. Let Ω be a bounded open subset of RN+1 and z ∂Ω. If
0
∈
∞ |Ωck(z0)| =+
k=1 λQQ+2klogk ∞
X
then z is -regular. In particular, the -regularity of z is ensured if Ω has the exterior
0 0
L L
-cone property at z .
0
L
WIENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 3
If E is a subset of either RN or RN+1, E stands for the relative Lebesgue measure.
| |
Moreover, Q is the homogeneous dimension recalled in Section 2, and the -cone property
L
will be defined precisely in Section 7. We just mention here that it is a natural adaptation
of the parabolic cone condition to the homogeneities of the operator .
L
Before proceeding, we would like to comment on Theorem 1.1 and Corollary 1.2.
A boundary point regularity test for the heat equation involving infinite sum of (caloric)
potentials was showed by Landis in [12]. A similar test for a Kolmogorov equation in R3
was obtained by Scornazzani in [14]. Our Theorem 1.1 contains, extends, and improves
the criterion in [14]. The Wiener test for the heat equation was proved by Evans and
Gariepy in [3]. The extension of such a criterion to the Kolmogorov operators (1.1) is
an open, and seemingly difficult, problem. Our Corollary 1.2, which is a straightforward
consequenceofTheorem1.1,isaWiener-typetestgivingnecessaryandsufficientconditions
which look “almost the same”. As a matter of fact, in Theorem 1.1 we have considered the
-potentials ofthe compactsets Ωc(z ) which arebuilt by the difference of two consecutive
L k 0
super-levelsets of Γ(z , ). These level sets correspondwith the sequence of values λ klogk.
0 −
·
The exact analogue of the Evans-Gariepy criterion would have required the sequence with
integer exponents λ k. The presence of the logarithmic term, which makes the growth of
−
the exponents slightly superlinear, is crucial for our proof of Theorem 1.1. Moreover, such
presence is also the responsible for the non-equivalence of the necessary and the sufficient
conditioninCorollary1.2. Tocompleteourhistoricalcomments,wementionthatapotential
analysis for Kolmogorovoperators of the kind (1.1) first appeared in [14], in [4], and in [9].
We alsomentionthatthe cone criterioncontainedinCorollary1.3hasbeenrecentlyproved
in [6], where such a boundary regularity test has been showed for classes of operators more
general than (1.1). For further bibliographical notes concerning Wiener-type tests for both
classical and degenerate operators, we refer the reader to [10].
The paper is organized as follows. In Section 2 we show some structural properties of
L
and fix some notations. Section 3 is devoted to the potential theory for , while in Section
L
4acrucialestimateoftheratiobetweenthe fundamentalsolutionΓ attwodifferentpolesis
proved. In Section 5 the only if part of Theorem 1.1 is proved. The if part, the core of our
paper, is proved in Section 6, where the estimates of Section 4 play a crucial rˆole. Section
7 is devoted to the proof of Corollary 1.2 and Corollary 1.3.
2. Structural properties of
L
In [9, Section 1] it is provedthat the operator is left-translation invariantwith respect
to the Lie group K whose underlying manifold isLRN+1, endowed with the composition law
(x,t) (ξ,s)=(ξ+E(s)x,t+s).
◦
Furthermore, a fundamental solution for is given by
L
Γ(z,ζ)=Γ ζ 1 z for z,ζ RN+1,
−
◦ ∈
where,
(cid:0) (cid:1)
0 for t 0,
≤
Γ(z)=Γ(x,t)=

 √(4dπe)t−CN(/t2)exp −41 C−1(t)x,x −ttrB for t>0.
(cid:0) (cid:10) (cid:11) (cid:1)

4 A.E.KOGOJ,E.LANCONELLI,ANDG.TRALLI
We assume the operator to be homogeneous of degree two with respect to a group
L
of dilations. This last assumption, together with the hypoellipticity of , implies that the
matrices A and B take the following form with respect to some basis oLf RN (see again [9,
Section 1]):
A 0
(2.1) A= 0
0 0
(cid:20) (cid:21)
for some p p symmetric and positive definite matrix A (p N), and
0 0 0 0
× ≤
0 0 ... 0 0
B 0 0 0 0
1
 
0 B ... 0 0
(2.2) B = 2 ,
 ... ... ... ... ...
 
 0 0 ... B 0
 n 
 
where B is a p p block with rank p (j = 1,2,...,n), p p ... p 1 and
j j 1 j j 0 1 n
− × ≥ ≥ ≥ ≥
p +p +...+p =N. ForsuchachoicewehavetrB =0,andthe family ofautomorphisms
0 1 n
of K making homogeneous of degree two can be taken as
L
δr :RN+1 RN+1, δr(x,t) = δr(x(p0),x(p1),...,x(pn),t)
−→
:= rx(p0),r3x(p1),...,r2n+1x(pn),r2t ,
x(pi) Rpi, i=0,...,n, r >0. (cid:16) (cid:17)
∈
We denote by Q+2 (= p +3p +...+(2n+1)p +2) the homogeneous dimension of K
0 1 n
with respect to (δ ) . We explicitly remark that Q is the homogenous dimension of RN
r r>0
with respect to the dilations
Dr :RN RN, Dr(x)= rx(p0),r3x(p1),...,r2n+1x(pn) .
−→
(cid:16) (cid:17)
Under these notations, the matrix C(t) and the fundamental solution of with pole at
L
the origin can be written as follows ([9, Proposition 2.3], see also [7]):
C(t)=D C(1)D
√t √t
and
0 for t 0,
≤
Γ(x,t)=

 tcQN2 exp −41hC−1(1)D√1tx,D√1txi for t>0.
(cid:16) (cid:17)
TWherooubgsehrovuettthhaet Γpaipseδrr-wheomdeongoenteeobuys of dthegereEeu−clQid.ean norms in RN, Rpk or R. We also
denote, for x RN, |·|
∈
1
x2 := C 1(1)x,x .
| |C 4 −
For all x RN, we have (cid:10) (cid:11)
∈
(2.3) E(1)x2 σ2 x2
| |C ≥ C| |
WIENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 5
where 4σ2 is the smallest eigenvalue of the positive definite matrix ET(1)C 1(1)E(1). We
C −
recallthatthe homogeneousnorm :RN R+ isaD -homogeneousfunctionofdegree
λ
k·k −→
1 defined as follows
n 1
x = x(pi) 2i+1 , for x= x(p0),...,x(pn) Rp0 ...Rpn =RN.
k k ∈ ×
Xi=0(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
We call homogeneo(cid:12)us cy(cid:12)linder of radius r >0 centered at 0 the set
:= x RN : x r t R : t r2 =δ ( ),
r r 1
C ∈ k k≤ × ∈ | |≤ C
and define r(z0):=z0(cid:8) r. (cid:9) (cid:8) (cid:9)
C ◦C
Remark 2.1. The norms and can be compared as follows
k·k |·|
(2.4) σmin x , x 2n1+1 x (n+1)max x , x 2n1+1 x RN,
| | | | ≤k k≤ | | | | ∀ ∈
n o n o
where σ =min x .
x=1
Indeed, on one|s|idekweksimply have
n
x x 2i1+1 (n+1)max x , x 2n1+1 x RN.
k k≤ | | ≤ | | | | ∀ ∈
Xi=0 n o
On the other hand, for any x=0, we get
6
x n x(pi) 2i1+1 n x (pi) 2i1+1 x
k k = = σ.
minn|x|,|x|2n1+1o ≥Xi=0 (cid:12)(cid:12) |x|2(cid:12)(cid:12)i1+1 Xi=0(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)|x|(cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)|x|(cid:13)(cid:13)(cid:13)(cid:13)≥
3. Some recalls from Potential Theory for :
L
-potentials and -capacity
L L
We briefly collect here some notions and results from Potential Theory applied to the
operator .
For eveLry open set Ω RN+1 we denote
⊆
(Ω):= u C (Ω) u=0 .
∞
L { ∈ | L }
and we call -harmonic in Ω the functions in (Ω).
L L
We say that a bounded open set V Ω is -regular if for every continuous function
ϕ:∂V R, there exists a unique funct⊆ion, hVLin (V), continuous in V, such that
−→ ϕ L
hV =ϕ.
ϕ|∂V
Moreover,if ϕ 0 then hV 0 by the minimum principle.
≥ ϕ ≥
A function u:Ω ] , ] is called -superharmonic in Ω if
−→ −∞ ∞ L
(i) u is lower semi-continuous and u< in a dense subset of Ω;
(ii) for every regular set V, V Ω, and∞for every ϕ C(∂V,R), ϕ u , it follows
∂V
⊆ ∈ ≤ |
u hV in V.
≥ ϕ
Wewilldenoteby (Ω)thefamilyofthe -superharmonicfunctionsinΩ. Sincetheoperator
endows RN+1 wLith a structure of βL-harmonic space satisfying the Doob convergence
L
6 A.E.KOGOJ,E.LANCONELLI,ANDG.TRALLI
property (see [13, 2, 6]), by the Wiener resolutivity theorem, for every f C(∂Ω), the
∈
Dirichlet problem
u=0 in Ω
L
(u∂Ω =f
|
has a generalized solution in the sense of Perron–Wiener–Bauer–Brelot given by
HΩ :=inf u (Ω) liminfu(z) f(ζ) ζ ∂Ω .
f { ∈L | Ω z ζ ≥ ∀ ∈ }
∋ →
The function HΩ is C (Ω) and satisfies u = 0 in Ω in the classical sense. However, it
f ∞ L
is not true, in general, that HΩ continuously takes the boundary values prescribed by f. A
f
point z ∂Ω such that
0
∈
lim HΩ(z)=f(z ) for every f C(∂Ω)
Ω∋z→z0 f 0 ∈
is called -regular for Ω.
For ouLr regularity criteria we still need a few more definitions. We denote by (RN+1)
the collection of all nonnegative Radon measure on RN+1 and we call M
Γ (z):= Γ(z,ζ) dµ(ζ), z RN+1,
µ
ZRN+1 ∈
the -potential of µ.
IfLF is a compact set of RN+1 and (F) is the collection of all nonnegative Radon
measure on RN+1 with support in F, theM-capacity of F is defined as
L
cap(F):=sup µ(RN+1) µ (F), Γ 1 on RN+1 .
µ
{ | ∈M ≤ }
We list some properties ofthe -capacitiescap. For every F, F andF compactsubsets of
1 2
RN+1, we have: L
(i) cap(F)< ;
∞
(ii) if F F , then cap(F ) cap(F );
1 2 1 2
⊆ ≤
(iii) cap(F F ) cap(F )+cap(F );
1 2 1 2
(iv) cap(z ∪F)=≤cap(F) for every z RN+1;
0 0
◦ ∈
(v) cap(δ (F))=rQcap(F) for every r >0;
r
(vi) if F =A τ for some compact set A RN, then cap(F)= A;
(vii) if F RN×{[a},b], then we have ⊂ | |
⊂ ×
F
(3.1) cap(F) | | .
≥ b a
−
Theproperties(i) (v)arequitestandard,andtheyfollowfromthefeaturesofΓ. Wewant
−
to spendfew wordsonthe lasttwoproperties. Property(vi) wasprovedin[8, Proposizione
5.1]inthecaseoftheheatoperator,namelywiththecapacitybuildontheGauss-Weierstrass
kernel. It can be proved verbatim proceeding in our situation: the main tools are the facts
that Γ has integral 1 over RN, and it reproduces the solutions of the Cauchy problems.
Property(vii) appears to be new even in the classicalparabolic case(at least to the best of
our knowledge), and it can be deduced readily from (vi). As a matter of fact, if a compact
set F lies in a strip RN [a,b], we have
×
b b b
(b a)cap(F)= cap(F)dτ cap(F t=τ )dτ = F t=τ dτ = F .
− ≥ ∩{ } | ∩{ }| | |
Za Za Za
WIENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 7
Thelastnotionsweneedaretheonesofreduced functionandofbalayageof1onF. They
are respectively defined by
W :=inf v v (RN+1), v 0 in RN+1, v 1 in F ,
F
{ | ∈L ≥ ≥ }
and
V (z)=liminfW (ζ), z RN+1.
F F
ζ z ∈
−→
FromgeneralbalayagetheorywehavethatV islessorequalthan1everywhere,identically
F
1 in the interior of F, it vanishes at infinity, is a superharmonic function on RN+1 and
harmonic on RN+1 ∂F. Furthermore, the following properties will be useful for us. Let
F,F1,F2 be compac\tsubsets of RN+1, andlet (Fn)n N be a sequence of compactsubsets of
RN+1, we have: ∈
(i) if F F RN+1, then V V ;
1 ⊆ 2 ⊆ F1 ≤ F2
(ii) V V +V ;
(iii) ifFF1∪⊆F2 ≤n FN1Fn, tFh2en VF ≤ ∞n=1VFn.
∈
The first property is a consequence of the definition of balayage; for the second and the
S P
third one we referrespectively to [1, Proposition5.3.1]and[1,Theorem4.2.2andCorollary
4.2.2].
Now, following the same lines of the proof of [8, Teorema 1.1],we have the existence of a
unique measure µ (F) such that
F
∈M
(3.2) V (z)=Γ (z)= Γ(z,ζ)dµ (ζ) z RN+1,
F µF ZRN+1 F ∀ ∈
and
µ (RN+1)=cap(F).
F
V is also called the -equilibrium potential of F and µ the -equilibrium measure of F.
F F
L L
TheproofofthisfactreliesonthegoodbehaviorofΓ,arepresentationformulaofRiesz-type
for -superharmonic functions proved in [2, Theorem 5.1], and a Maximum Principle for
L L
(see [2, Proposition 2.3]).
Fix now a bounded open set Ω compactly contained in RN+1, and z = (x ,t ) ∂Ω.
0 0 0
∈
Let us denote by
G = (x,t) (z )rΩ : t t .
r r 0 0
{ ∈C ≤ }
From general balayage theory and proceeding, e.g., as in [11, Theorem 4.6], we can charac-
terize the regularity of the boundary point of Ω by the following condition:
the point z ∂Ω is -regular if and only if
0
∈ L
(3.3) limV (z )>0.
r 0 Gr 0
→
4. A crucial estimate
We start by recalling the following identity, whose proof can be found in [9, Remark 2.1]
(see also [7]),
(4.1) E(λ2s)D =D E(s) λ>0, s R.
λ λ
∀ ∀ ∈
In what follows we will need the following lemma.
Lemma 4.1. For 0>t>τ we have the following matrix inequality
ET(t)C 1(t τ)E(t) C 1( τ).
− −
− ≥ −
8 A.E.KOGOJ,E.LANCONELLI,ANDG.TRALLI
Proof. Since for symmetric positive definite matrices we have
M M M 1 M 1
1 ≤ 2 ⇒ 1− ≥ 2−
(see [5, Corollary 7.7.4]) and recalling that E 1(t)=E( t), it is enough to prove that
−
−
(4.2) E( t)C(t τ)ET( t) C( τ).
− − − ≤ −
From the very definition of the matrix C we get
t τ t τ
E( t)C(t τ)ET( t) = etB − e sBAe sBT ds etBT = − e(t s)BAe(t s)BT ds
− − − −
− − −
(cid:18)Z0 (cid:19) Z0
τ
= − e σBAe σBT dσ.
− −
Z−t
Since τ > t>0 and A is nonnegative definite, we have
− −
τ τ
− e σBAe σBT dσ − e σBAe σBTdσ =C( τ)
− − − −
≤ −
Z−t Z0
which proves (4.2) and the lemma. (cid:3)
A crucial role in the proof of our main theorem will be played by the ratio Γ(z,ζ), for
Γ(0,ζ)
z =(x,t) and ζ =(ξ,τ) with 0>t>τ. We use the following notations
t
µ= − (0,1), M(z)= D x , M(ζ)= D ξ .
1 1
τ ∈ √ t √ τ
− (cid:12) − (cid:12) (cid:12) − (cid:12)
Lemma 4.2. There exists a positive constan(cid:12)t C such(cid:12) that, for any z(cid:12)=(x,t)(cid:12),ζ =(ξ,τ) with
(cid:12) (cid:12) (cid:12) (cid:12)
0>t>τ and µ min 1, σ2 , we have
≤ {2 (n+1)2}
Q
Γ(z,ζ) 1 2
eC√µM(z)M(ζ).
Γ(0,ζ) ≤ 1 µ
(cid:18) − (cid:19)
Proof. In our notations we can write
(cid:12) (cid:12)2
Γ(z,ζ) = (t−τ)−Q2e−(cid:12)(cid:12)(cid:12)D√t1−τ(x−E(t−τ)ξ)(cid:12)(cid:12)(cid:12)C
Γ(0,ζ) (cid:12) (cid:12)2
(−τ)−Q2e−(cid:12)(cid:12)(cid:12)D√−1τ(E(−τ)ξ)(cid:12)(cid:12)(cid:12)C
Q (cid:12) (cid:12)2 (cid:12) (cid:12)2
= 1 1 µ 2 e(cid:12)(cid:12)(cid:12)D√−1τ(E(−τ)ξ)(cid:12)(cid:12)(cid:12)C−(cid:12)(cid:12)(cid:12)D√t1−τ(x−E(t−τ)ξ)(cid:12)(cid:12)(cid:12)C.
(cid:18) − (cid:19)
Let us deal with the exponential term
2 2
(4.3) D (E( τ)ξ) D (x E(t τ)ξ) =
1 1
√ τ − C − √t τ − − C
(cid:12)1 − (cid:12) (cid:12) − 1 (cid:12)
= (cid:12) C 1( τ)E( (cid:12)τ)ξ,E(cid:12)( τ)ξ C 1(t τ(cid:12))(x E(t τ)ξ),(x E(t τ)ξ) .
(cid:12) − (cid:12) (cid:12) − (cid:12)
4 − − − − 4 − − − − −
Lemma 4.1(cid:10)says in particular that we(cid:11)have(cid:10) (cid:11)
C 1( τ)E( τ)ξ,E( τ)ξ C 1(t τ)E(t τ)ξ,E(t τ)ξ 0.
− −
− − − − − − − ≤
(cid:10) (cid:11) (cid:10) (cid:11)
WIENER-LANDIS CRITERION FOR KOLMOGOROV-TYPE OPERATORS 9
Using this in (4.3) we get
2 2
(4.4) D (E( τ)ξ) D (x E(t τ)ξ)
1 1
√ τ − C − √t τ − − C ≤
(cid:12) 1 − (cid:12) (cid:12) 1 − (cid:12) 1
(cid:12) C 1(t τ)x(cid:12),x +(cid:12) C 1(t τ)x,E(t (cid:12)τ)ξ C 1(t τ)x,E(t τ)ξ
(cid:12) − (cid:12) (cid:12) − (cid:12) −
≤ −4 − 2 − − ≤ 2 − −
1 (cid:10) (cid:11) (cid:10) (cid:11) 1(cid:10) (cid:11)
C 1(t τ)x,x C 1(t τ)E(t τ)ξ,E(t τ)ξ 2 .
− −
≤ 2 − − − −
We are goi(cid:0)n(cid:10)g to bound C(cid:11)(cid:10)1(t τ)x,x and C 1(t τ)E((cid:11)t(cid:1) τ)ξ,E(t τ)ξ separately.
− −
− − − −
We have
(cid:10) (cid:11) (cid:10) (cid:11)
1 1
C 1(t τ)x,x = C 1 1 D x,D x C 1 1 M2(z),
− − 1 1 −
− µ − √ t √ t ≤ µ −
(cid:28) (cid:18) (cid:19) − − (cid:29) (cid:13) (cid:18) (cid:19)(cid:13)
(cid:10) (cid:11) (cid:13) (cid:13)
where A stands for the operator norm of a matrix A (i.(cid:13)e. its biggest(cid:13)eigenvalue for
k k (cid:13) (cid:13)
symmetric matrices). By (2.4), for any vector v with v =1 we get
| |
1 1 n+1 1 n+1
min D√µv , D√µv 2n+1 ≤ σ√µkvk≤ σ √µmax |v|,|v|2n+1 = σ √µ.
From µn≤(cid:12)(cid:12)(n+σ21)2(cid:12)(cid:12) w(cid:12)(cid:12)e then(cid:12)(cid:12)deduoce D√µv ≤ n+σ1√µ. Hence, snince µ is alsoo less than 12,
C 1 1 1 v,v = C 1((cid:12)(cid:12)1 µ)(cid:12)(cid:12)D v,D v C 1(1 µ) D v 2
− µ − − − √µ √µ ≤ − − √µ
(cid:28) (cid:18) (cid:19) (cid:29)
(cid:10)(n+1)2 C−1(1 µ) µ(cid:11) (n(cid:13)(cid:13)+1)2 C−1(cid:13)(cid:13)(cid:12)(cid:12)1 µ(cid:12)(cid:12) v =1.
≤ σ2 − ≤ σ2 2 ∀| |
(cid:13) (cid:18) (cid:19)(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
This gives (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13)
(n+1)2 1
(4.5) C−1(t τ)x,x C−1 µM2(z).
− ≤ σ2 2
(cid:13) (cid:18) (cid:19)(cid:13)
(cid:10) (cid:11) (cid:13) (cid:13)
On the other hand, by the commutation property(cid:13)(4.1), we g(cid:13)et
(cid:13) (cid:13)
C 1(t τ)E(t τ)ξ,E(t τ)ξ = C 1(1 µ)D E(t τ)ξ,D E(t τ)ξ
− − 1 1
− − − − √ τ − √ τ −
− −
(cid:10) 2(cid:11) D 2 E
C 1(1 µ) D E(t τ)ξ = C 1(1 µ) E(1 µ)D ξ
− 1 − 1
≤ − √ τ − − − √ τ
≤ (cid:13)(cid:13)C−1(1−µ)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)ET−(1−µ)E(1−(cid:12)(cid:12)(cid:12) µ)(cid:13)(cid:13)M2(ζ) (cid:13)(cid:13)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)
1
(cid:13)(cid:13)C−1 (cid:13)(cid:13)E(cid:13)(cid:13)T(1 µ)E(1 µ) M(cid:13)(cid:13) 2(ζ).
≤ 2 − −
(cid:13) (cid:18) (cid:19)(cid:13)
(cid:13) (cid:13)(cid:13) (cid:13)
Since 0(cid:13)< µ 1,(cid:13)t(cid:13)he term ET(1 µ)(cid:13)E(1 µ) is bounded from above by a universal
(cid:13) ≤ 2(cid:13) − −
constant C2. Thus we have
0 (cid:13) (cid:13)
(cid:13) (cid:13)
1
(4.6) C 1(t τ)E(t τ)ξ,E(t τ)ξ C2 C 1 M2(ζ).
− − − − ≤ 0 − 2
(cid:13) (cid:18) (cid:19)(cid:13)
(cid:10) (cid:11) (cid:13) (cid:13)
Plugging (4.5) and (4.6) in (4.4), we get (cid:13) (cid:13)
(cid:13) (cid:13)
2 2 C n+1 1
D√1τ (E(−τ)ξ) C − D√t1 τ (x−E(t−τ)ξ) C ≤ 20 σ C−1 2 √µM(z)M(ζ).
(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13) (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)
10 A.E.KOGOJ,E.LANCONELLI,ANDG.TRALLI
Therefore
Q
Γ(z,ζ) 1 2
eC√µM(z)M(ζ)
Γ(0,ζ) ≤ 1 µ
(cid:18) − (cid:19)
with C = C20n+σ1 C−1 21 . (cid:3)
(cid:13) (cid:0) (cid:1)(cid:13)
(cid:13) 5. N(cid:13)ecessary condition for regularity
The characterization in (3.3), together with the following lemma, will give the necessity
of (1.3) in Theorem 1.1.
Lemma 5.1. For every fixed p N, let us split the set G as follows
r
∈
Gr =Gpr ∪G∗rp,
where
plogp
1
Gp = z G Γ(z ,z) z ,
r ( ∈ r | 0 ≥(cid:18)λ(cid:19) )∪{ 0}
plogp
1
and G p = z G Γ(z ,z) .
∗r ( ∈ r | 0 ≤(cid:18)λ(cid:19) )
Then,
rlim0VGr(z0)=rlim0VGpr(z0).
−→ −→
Proof. From the monotonicity and subadditivity properties of the balayage,we have
VGpr(z0)≤VGr(z0)≤VGpr(z0)+VG∗rp(z0).
Furthermore, by (3.2),
1 plogp
VG∗rp(z0)≤ λ cap(G∗rp).
(cid:18) (cid:19)
On the other hand, from the monotonicity and homogeneity properties of the capacities, it
follows
cap(G p) cap( (z ))=cap(z δ ( ))=rQcap( (z )).
∗r ≤ Cr 0 0◦ r C1 C1 0
Hence cap(G p) goes to zero as r goes to zero. This proves the lemma. (cid:3)
∗r
Proof of necessary condition in Theorem 1.1. Assume
∞
V (z )<+ .
Ωck(z0) 0 ∞
k=1
X
We aregoing to provethe nonregularityofthe boundarypoint z . The assumptionimplies
0
that for every ε>0, there exists p N such that
∈
∞
V (z )<ε.
Ωck(z0) 0
k=p
X
On the other hand, with the notations of the previous lemma, for any positive r
∞
Gp Ωc(z ),
r ⊆ k 0
k=p
[