Some properties for a subfunction associated with the stationary Schrödinger operator in a cone

Long Pinhong, Deng, Deng Guantie

Description

For a subfunction u associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we correct Theorem 1 in (Qiao and Deng in Glasg. Math. J. 53(3):599-610, 2011). Then by the theorem we generalize some theorems of Phragmén-Lindelöf type for a subfunction in a cone. Meanwhile, we obtain some results about the existence of solutions of the Dirichlet problem associated with the stationary Schrödinger operator in a cone and about the type of their uniqueness. MSC: 31B05.

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• Coné

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

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Long and DengJournal of Inequalities and Applications2012,2012:295 http://www.journalofinequalitiesandapplications.com/content/2012/1/295

R E S E A R C HOpen Access Some properties for a subfunction associated with the stationary Schrödinger operator in a cone * Pinhong Long and Guantie Deng

* Correspondence: [e-mail] School of Mathematical Science, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, P.R. China

Abstract For a subfunctionuassociated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we correct Theorem 1 in (Qiao and Deng in Glasg. Math. J. 53(3):599-610, 2011). Then by the theorem we generalize some theorems of Phragmén-Lindelöf type for a subfunction in a cone. Meanwhile, we obtain some results about the existence of solutions of the Dirichlet problem associated with the stationary Schrödinger operator in a cone and about the type of their uniqueness. MSC:Primary 31B05 Keywords:stationary Schrödinger operator; Poissona-integral; subfunction; cone

1 Introductionand main results To begin with, let us agree to some basic conventions. As usual, letSbe an open set in n n R(n≥), whereRis then-dimensional Euclidean space. The boundary and the closure ofSare denoted by∂SandS, respectively. LetP= (X,x n ), whereX= (x  ,x  , . . . ,x n– ), and let|P|be the Euclidean norm ofPand|P–Q|be the Euclidean distance of two points n n– PandQinR. The unit sphere and the upper half unit sphere are denoted bySand n–n– S, respectively. For simplicity, the point (,) onSand the set{; (,)∈}for a + n–n– set,⊂Sare often identified withand, respectively. For⊂R + and⊂S, n n the set{(r,)∈R;r∈, (,)∈}inRis simply denoted by×. In particular, the n–n ceR + ×S={ n )∈R;x n > }will b half-spa + (X,xe denoted byT n . ByC n (), we denote n n– the setR + ×inRwith the domainonSand call it a cone. For an intervalI⊂ n– R + and⊂S, setC n (;I) =I×,S n (;I) =I×∂andC n (;r) =C n ()∩S r . By S n () we denoteS n (; (, +∞)), which is∂C n () –{O}. Furthermore, we denote bydS r the (n– )-dimensional volume elements induced by the Euclidean metric onS r . ForP∈ n n Randr> , letB(P,r) denote the open ball with center atPand radiusrinR, then S r =∂B(O,r). We introduce the system of spherical coordinates (r,),= (θ  ,θ  , . . . ,θ n– ) forP= n (X,x n ) inRvia the following formulas:

n–  x  =rsinθ j (n≥),x n =rcosθ  j= ©2012 Long and Deng; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Long and DengJournal of Inequalities and Applications2012, 2012:295 http://www.journalofinequalitiesandapplications.com/content/2012/1/295

and ifn≥,

k–  x n–k+ =rcosθ k sinθ j (≤k≤n– ), j=

ππ where ≤r<∞, ≤θ j ≤π(≤j≤n– ) and –≤θ n– ≤.   Relative to the system of spherical coordinates, the Laplace operatormay be written

 * n– ∂ ∂ + ,= +   r∂r∂r r

* where the explicit form of the Beltrami operatoris given by Azarin (see []). n For an arbitrary domainDinR,A D denotes the class of non-negative radial potentials b a(P) (i.e., ≤a(P) =a(r) forP= (r,)∈D) such thata∈L(D) with someb>n/ ifn≥ loc and withb=  ifn=  orn= . Ifa∈A D , the stationary Schrödinger operator with a potentiala(∙),

L a = –+a(∙)I,

(.)

∞ can be extended in the usual way from the spaceC(D) to an essentially self-adjoint oper-   ator onL(D), whereis the Laplace operator andIis the identical operator (see Reed and a a Simon [], Chapter ). Furthermore, L a has a Greena-functionG(∙,∙). HereG(∙,∙) is D D a positive onDand its inner normaive∂G(∙,Q)/∂n Q ≥, where l derivat D ∂/∂n Q denotes the a differentiation atQalong the inward normal intoD. We denote this derivative byPI(∙,∙), D which is called the Poissona-kernel with respect toD. There is an inequality between the a  Greena-functionG(∙,∙) and that of the Laplacian, hereafter denoted byG(∙,∙). It is well D D known that for any potentiala(∙)≥,

a G(∙,∙)≤G(∙,∙). D D

(.)

n The inverse inequality is much more elaborate whenDis a bounded domain inR. For a n bounded domainDinR, Cranston, Fabes and Zhao (see [], the casen=  is implicitly contained in []) have proved

a G(∙,∙)≥M(D)G(∙,∙), D D

(.)

whereM(D) =M(D,a) is positive and independent of points inD. Ifa= , obviously M(D)≡. Suppose that a functionu≡–∞is upper semi-continuous inD.u∈[–∞, +∞) is called a subfunction of the Schrödinger operator L a if the generalized mean-value inequality

 a ∂G(P,Q) B(P,ρ) u(P)≤u(Q)dσ(Q) ∂n Q S(P,ρ)

(.)

is satisfied at each pointP∈Dwith  <ρ<inf Q∈∂D |P–Q|, whereS(P,ρ) =∂B(P,ρ), a G(P,∙) is the Greena-function of L a inB(P,ρ) anddσ(∙) is the surface area element B(P,ρ) onS(P,ρ).

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