R E S E A R C H
Open Access
Minimally thin sets associated with the
stationary Schrödinger operator
Tao Zhao
**Correspondence: [email protected]
College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450000, P.R. China
Abstract
This paper gives some new criteria fora-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by Long-Gao-Deng.
MSC: 31B05; 31B10
Keywords: minimally thin set; Schrödinger operator; Greena-potential
1 Introduction and results
LetRandR+ be the set of all real numbers and the set of all positive real numbers, re-spectively. We denote byRn(n≥) then-dimensional Euclidean space. A point inRnis denoted byP= (X,xn),X= (x,x, . . . ,xn–). The Euclidean distance between two pointsP andQinRnis denoted by|P–Q|. Also|P–O|withOthe origin ofRnis simply denoted
by|P|. The boundary and the closure of a setSinRnare denoted by∂SandS, respectively.
We introduce a system of spherical coordinates (r,),= (θ,θ, . . . ,θn–), inRnwhich are related to Cartesian coordinates (x,x, . . . ,xn–,xn) byxn=rcosθ.
LetDbe an arbitrary domain inRnandA
adenote the class of nonnegative radial
po-tentialsa(P),i.e.≤a(P) =a(r),P= (r,)∈D, such thata∈Lbloc(D) with someb>n/ if
n≥ and withb= ifn= orn= .
Ifa∈Aa, then the stationary Schrödinger operator
Scha= –+a(P)I= ,
whereis the Laplace operator andIis the identical operator, can be extended in the usual way from the spaceC∞(D) to an essentially self-adjoint operator onL(D) (see [, Ch. ]). We will denote itScha as well. This last one has a Greena-functionGaD(P,Q).
HereGaD(P,Q) is positive onDand its inner normal derivative∂GaD(P,Q)/∂nQ≥, where
∂/∂nQdenotes differentiation atQalong the inward normal intoD.
We call a functionu≡–∞that is upper semi-continuous inDa subfunction with re-spect to the Schrödinger operatorSchaif it values belong to the interval [–∞,∞) and at
each pointP∈Dwith <r<r(P) the generalized mean-value inequality (see [])
u(P)≤
S(P,r)
u(Q)∂G
a
B(P,r)(P,Q)
∂nQ
dσ(Q)
is satisfied, whereGBa(P,r)(P,Q) is the Greena-function ofScha inB(P,r) anddσ(Q) is a
surface measure on the sphereS(P,r) =∂B(P,r). If –uis a subfunction, then we callua superfunctions (with respect to the Schrödinger operatorScha). If a functionuis both
sub-function and supersub-function, it is, clearly, continuous and is called ana-harmonic function (with respect to the Schrödinger operatorScha).
The unit sphere and the upper half unit sphere in Rnare denoted bySn–andSn– + , respectively. For simplicity, a point (,) on Sn– and the set {; (,)∈} for a set
,⊂Sn–, are often identified withand, respectively. For two sets⊂R + and
⊂Sn–, the set {(r,)∈Rn;r∈, (,)∈}inRn is simply denoted by×. By
Cn(), we denote the setR+×inRnwith the domainonSn–. We call it a cone. We denote the setI×with an interval onRbyCn(;I).
From now on, we always assume D=Cn(). For the sake of brevity, we shall write
Ga(P,Q) instead ofGCan()(P,Q). Throughout this paper, letcdenote various positive
con-stants, because we do not need to specify them.
Letbe a domain onSn–with smooth boundary. Consider the Dirichlet problem ( n+λ)ϕ= on,
ϕ= on∂,
where nis the spherical part of the Laplace operatorn
n=
n–
r ∂ ∂r+
∂ ∂r +
n
r .
We denote the least positive eigenvalue of this boundary value problem byλand the nor-malized positive eigenfunction corresponding toλbyϕ(). In order to ensure the exis-tence ofλand a smoothϕ(), we put a rather strong assumption on: ifn≥, then
is aC,α-domain ( <α< ) onSn–surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g.see [, pp.-] for the definition ofC,α-domain).
For any (,)∈, we have (see [, pp.-])
c–rϕ()≤δ(P)≤crϕ(), ()
whereP= (r,)∈Cn() andδ(P) =dist(P,∂Cn()).
We study solutions of an ordinary differential equation,
–Q (r) –n–
r Q(r) +
λ r+a(r)
Q(r) = , <r<∞. ()
It is well known (see, for example, []) that if the potentiala∈Aa, then equation () has
a fundamental system of positive solutions {V,W} such thatV is nondecreasing with (see [])
≤V(+)≤V(r) ∞ asr→+∞,
andWis monotonically decreasing with
We will also consider the classBa, consisting of the potentialsa∈Aasuch that the finite
limitlimr→∞ra(r) =k∈[,∞) exists, and moreover,r–|ra(r) –k| ∈L(,∞). Ifa∈Ba,
then the (sub)superfunctions are continuous (see []). In the rest of this paper, we assume thata∈Baand we shall suppress this assumption for simplicity.
Denote
ι±k = –n±
(n– )+ (k+λ)
,
then the solutions to equation () have the asymptotic (see [])
c–rι+k≤V(r)≤crι+k, c–rι–k≤W(r)≤crι–k asr→ ∞. ()
It is well known that the Martin boundary ofCn() is the set∂Cn()∪ {∞}, each point
of which is a minimal Martin boundary point. ForP∈Cn() andQ∈∂Cn()∪ {∞}, the
Martin kernel can be defined byMa
(P,Q). If the reference pointPis chosen suitably, then we have
Ma(P,∞) =V(r)ϕ() and Ma(P,O) =cW(r)ϕ() ()
for anyP= (r,)∈Cn().
In [], Long-Gao-Deng introduce the notations ofa-thin (with respect to the Schrö-dinger operatorScha) at a point, a-polar set (with respect to the Schrödinger operator
Scha) anda-minimal thin sets at infinity (with respect to the Schrödinger operatorScha),
which generalized earlier notations obtained by Brelot and Miyamoto (see [, ]). A setH
inRnis said to bea-thin at a pointQif there is a fine neighborhoodEofQwhich does not intersectH\{Q}. OtherwiseHis said to be nota-thin atQonCn(). A setHinRnis called
a polar set if there is a superfunctionuon some open setEsuch thatH⊂ {P∈E;u(P) =∞}. A subsetHofCn() is said to bea-minimal thin atQ∈∂Cn()∪ {∞}onCn(), if there
exists a pointP∈Cn() such that
ˆ
RHMa
(·,Q)(P)=M
a
(P,Q),
whereRˆH
Ma(·,Q)is the regularized reduced function ofMa(·,Q) relative toH(with respect to the Schrödinger operatorScha).
LetHbe a bounded subset ofCn(). ThenRˆHMa
(·,∞)(P) is bounded onCn() and hence
the greatest a-harmonic minorant ofRˆH
Ma(·,∞) is zero. When byG a
μ(P) we denote the Greena-potential with a positive measureμonCn(), we see from the Riesz
decomposi-tion theorem (see [, Theorem ]) that there exists a unique positive measureλaHonCn()
such that (see [, p.])
ˆ
RHMa
(·,∞)(P) =G
a
λ
a H(P)
for anyP∈Cn() andλaHis concentrated onIH, where
IH=
P∈Cn();His nota-thin atP
The Green a-energyγa(H) (with respect to the Schrödinger operatorScha) ofλaH is
defined by
γa(H) =
Cn()
GaλaHdλaH.
Also, we can define a measureσa onCn()
σa(H) =
H
Ma(P,∞)
δ(P)
dP.
Recently, Long-Gao-Deng (see [, Theorem .]) gave a criterion that characterizes
a-minimally thin sets at infinity in a cone.
Theorem A A subset H of Cn()is a-minimally thin at infinity on Cn()if and only if
∞
j=
γa(Hj)W j
V– j<∞,
where Hj=H∩Cn(; [j, j+))and j= , , , . . . .
In this paper, we shall obtain a series of new criteria fora-minimally thin sets at infinity onCn(), which complemented Theorem A by the way completely different from theirs.
Our results are essentially based on Ren and Su (see [, ]). First we have the following.
Theorem The following statements are equivalent.
(I) A subsetHofCn()isa-minimally thin at infinity onCn().
(II) There exists a positive superfunctionv(P)onCn()such that
inf
P∈Cn()
v(P)
Ma
(P,∞)
= ()
and
H⊂P∈Cn();v(P)≥Ma(P,∞)
.
(III) There exists a positive superfunctionv(P)onCn()such that even if
v(P)≥cMa(P,∞)for anyP∈H,there existsP∈Cn()satisfying
v(P) <cMa(P,∞).
Next we shall state Theorem , which is the main result in this paper.
Theorem If a subset H of Cn()is a-minimally thin at infinity on Cn(),then we have
H
dP
2 Lemmas
In our discussions, the following estimate for the Greena-potentialGa(P,Q) is funda-mental, as follows from [].
Lemma
c–V(r)W(t)ϕ()ϕ()≤Ga(P,Q)≤cV(r)W(t)ϕ()ϕ()
for any P= (r,)∈Cn()and any Q= (t,)∈Cn()satisfying t≥r.
Lemma If H is a bounded Borel subset of Cn(),then
σa(H)≤cγa(H).
Proof For anyP∈Rn\C
n() and any positive numberr> , there exists a positive constant
csuch that
Cap P+r–(Q–P)∈Rn;Q∈B(P,r)∩ Rn\Cn()
≥c
from [, p.], whereCapdenotes the Newtonian capacity. Then there exists a positive constantcdepending only oncandnsuch that
Cn()
(P)
δ(P)
dP≤c
Cn()
∇(P)dP ()
for every(P)∈C∞ (Cn()) (see [, Theorem ]).
It is well known that the Greena-energy also can be represented as (see [, p.])
γa(H) =
Cn()
∇GaλaH(P)
dP. ()
From equation () and Lemma we have
Cn()
GaλaH(P)
δ(P)
dP<∞. ()
From equations () and () we obtainGaλaH(P)∈, where
=
f ∈Lloc Cn()
;∇f ∈L Cn()
,δ–f∈L Cn()
equipped with the norm
f= ∇f
L(Cn())+δ–f
L(Cn())
,
and furtherGa
λaH(P)∈, where denotes the closure ofC∞(Cn()) in. Thus we obtain from equation () (see [, p.])
Cn()
GaλaH(P)
δ(P)
dP≤c
Cn()
SinceGaλaH=Ma(·,∞) quasi everywhere onHand hence a.e. onH, we have from equa-tion ()
γa(H)≥c–
Cn()
Ga
λaH(P)
δ(P)
dP
≥c–
Cn()
Ma
(P,∞)
δ(P)
dP
=c–σa(H),
which gives the conclusion of Lemma .
3 Proof of Theorem 1
We shall show that (II) follows from (I). Since
ˆ
RHMja
(·,∞)(Q) =M
a
(Q,∞)
for anyQ∈IHjandλHj is concentrated onIHj, we have
γa(Hj) =
IHj
Ma(Q,∞)dλaHj(Q)
≥V j
IHj
ϕ()dλaH j(Q)
for anyQ= (t,)∈Cn() and hence from Lemma
ˆ
RHMja
(·,∞)(P)≤cV(r)ϕ()
IHj
W(t)ϕ()dλaH j(Q)
≤cV(r)ϕ()W jV– jγa(Hj) ()
for anyP= (r,)∈Cn() and any integerjsatisfying j≥r.
If we define a measureμonCn() by
μ= ∞
j=
λaH j,
then
Gaμ(P) = ∞
j=
ˆ
RHMja (·,∞)(P).
From equation (), (I), and Theorem A, we know thatGaμ(P) is a finite superfunction onCn() and
Gaμ(P)≥ ˆR
Hj
for anyP= (r,)∈IHj (j= , , , , . . .) and from Lemma
Gaμ(P)≥cV(r)ϕ()
for anyP= (r,)∈Cn(; (, )) and
c=c–
Cn(;[r,∞))
W(t)ϕ()dμ(Q).
If we setH =∞j=–IHj, where
H–=H∩Cn ; (, )
, ()
andc=min{c, }, then
H ⊂P= (r,)∈Cn();Gaμ(P)≥cV(r)ϕ()
andH is equal toHexcept a polar setH. If we define a positive measureηonCn() such
thatGaμis identically +∞onHand define a measureνonCn() byν=c– (μ+η), then
H⊂P= (r,)∈Cn();Gaν(P)≥V(r)ϕ()
.
If we putv(P) =Ga
ν(P), then this shows thatv(P) is the function required in (II). Now we shall show that (III) follows from (II). Letv(P) be the function in (II). It follows thatv(P)≥Ma
(P,∞) for anyP∈H. On the other hand, from equation () we can find a pointP∈Cn() such thatv(P) <Ma(P,∞). Thereforev(P) satisfies (III) withc= .
Finally, we shall prove that (I) follows from (III). Letv(P) be the function in (III). If we put
inf
P∈Cn() v(P)
Ma(P,∞)
=c(∞,v)
and
u(P) =v(P) –c(∞,v)Ma(P,∞),
then we have
inf
P∈Cn() u(P)
Ma(P,∞) = ,
wherec(∞,v) is a positive constant depending only on∞andv. Since there existsP∈
Cn() satisfyingv(P) <cMa(P,∞), we note thatc>c(∞,v). Now we obtainu(P)≥ (c–c(∞,v))Ma(P,∞) for anyP∈H. Hence by a result of [, p.],H isa-minimally thin at infinity onCn() with respect to the Schrödinger operator, which is the statement
4 Proof of Theorem 2 First of all, we remark that
H
dP
( +|P|)n =
H–
dP
( +|P|)n+
∞
j=
Hj dP
( +|P|)n
≤ |H–|+ ∞
j=
–jn|Hj|, ()
whereH–is the set in equation () and|Hj|is then-dimensional Lebesgue measure ofHj.
We have from equations () and ()
σa(Hj) =
Hj
Ma
(P,∞)
δ(P)
dP
≥c
Hj
V(r)ϕ()
rϕ()
dP
≥c
Hj
rι+k–dP
≥c
Hj
j(ι+k–)dP
=cj(ι+k–)|Hj|.
By using Lemma , we obtain
γa(Hj)≥c–σa(Hj)≥cj(ι
+
k–)|Hj|. ()
IfHisa-minimally thin at infinity onCn(), then from Theorem A, equations (), (),
and (), we have
H
dP
( +|P|)n ≤ |H–|+c
∞
j=
j(ι+k–)|Hj|W jV– j
≤ |H–|+c ∞
j=
γa(Hj)W j
V– j <∞,
which is the conclusion of Theorem .
Competing interests
The author declares that there is no conflict of interests regarding the publication of this article.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102. The author would like to thank two anonymous referees for numerous insightful comments and suggestions, which have greatly improved the paper.
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