Martin kernel of Schrödinger operators with singular potentials
and applications to B.V.P. for linear elliptic equations

J 2022

Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations

GKIKAS, Konstantinos T. and Phuoc Tai NGUYEN

Basic information

Original name

Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations

Authors

GKIKAS, Konstantinos T. (300 Greece, guarantor) and Phuoc Tai NGUYEN (704 Viet Nam, belonging to the institution)

Edition

Calculus of Variations and Partial Differential Equations, Springer, 2022, 0944-2669

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Germany

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 2.100

RIV identification code

RIV/00216224:14310/22:00119376

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1007/s00526-021-02102-6

UT WoS

000717551300005

Keywords in English

Schrödinger operators; Singular elliptic equations; Green's functions; Boundary value problems for second-order elliptic equations

Abstract

V originále

Let \(\Omega \subset {\mathbb {R}}^N\) (\(N \ge 3\)) be a \(C^2\) bounded domain and \(\Sigma \subset \Omega \) be a compact, \(C^2\) submanifold in \({\mathbb {R}}^N\) without boundary, of dimension k with \(0\le k < N-2\). Denote \(d_\Sigma (x): = \mathrm {dist}\,(x,\Sigma )\) and \(L_\mu : = \Delta + \mu d_\Sigma ^{-2}\) in \(\Omega {\setminus } \Sigma \), \(\mu \in {\mathbb {R}}\). The optimal Hardy constant \(H:=(N-k-2)/2\) is deeply involved in the study of the Schrödinger operator \(L_\mu \). The Green kernel and Martin kernel of \(-L_\mu \) play an important role in the study of boundary value problems for nonhomogeneous linear equations involving \(-L_\mu \). If \(\mu \le H^2\) and the first eigenvalue of \(-L_\mu \) is positive then the existence of the Green kernel of \(-L_\mu \) is guaranteed by the existence of the associated heat kernel. In this paper, we construct the Martin kernel of \(-L_\mu \) and prove the Representation theory which ensures that any positive solution of the linear equation \(-L_\mu u = 0\) in \(\Omega {\setminus } \Sigma \) can be uniquely represented via this kernel. We also establish sharp, two-sided estimates for Green kernel and Martin kernel of \(-L_\mu \). We combine these results to derive the existence, uniqueness and a priori estimates of the solution to boundary value problems with measures for nonhomogeneous linear equations associated to \(-L_\mu \).

Links

GJ19-14413Y, research and development project

Name: Lineární a nelineární eliptické rovnice se singulárními daty a související problémy

Investor: Czech Science Foundation

Citovat

GKIKAS, Konstantinos T. and Phuoc Tai NGUYEN. Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations. Calculus of Variations and Partial Differential Equations. Springer, 2022, vol. 61, No 1, p. 1-36. ISSN 0944-2669. Available from: https://dx.doi.org/10.1007/s00526-021-02102-6.

@article{1806077,
   author = {Gkikas, Konstantinos T. and Nguyen, Phuoc Tai},
   article_number = {1},
   doi = {http://dx.doi.org/10.1007/s00526-021-02102-6},
   keywords = {Schrödinger operators; Singular elliptic equations; Green's functions; Boundary value problems for second-order elliptic equations},
   language = {eng},
   issn = {0944-2669},
   journal = {Calculus of Variations and Partial Differential Equations},
   title = {Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations},
   url = {https://link.springer.com/article/10.1007/s00526-021-02102-6},
   volume = {61},
   year = {2022}
}

TY  - JOUR
ID  - 1806077
AU  - Gkikas, Konstantinos T. - Nguyen, Phuoc Tai
PY  - 2022
TI  - Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations
JF  - Calculus of Variations and Partial Differential Equations
VL  - 61
IS  - 1
SP  - 1-36
EP  - 1-36
PB  - Springer
SN  - 09442669
KW  - Schrödinger operators
KW  - Singular elliptic equations
KW  - Green's functions
KW  - Boundary value problems for second-order elliptic equations
UR  - https://link.springer.com/article/10.1007/s00526-021-02102-6
N2  - Let \(\Omega \subset {\mathbb {R}}^N\) (\(N \ge 3\)) be a \(C^2\) bounded domain and \(\Sigma \subset \Omega \) be a compact, \(C^2\) submanifold in \({\mathbb {R}}^N\) without boundary, of dimension k with \(0\le k < N-2\). Denote \(d_\Sigma (x): = \mathrm {dist}\,(x,\Sigma )\) and \(L_\mu : = \Delta + \mu d_\Sigma ^{-2}\) in \(\Omega {\setminus } \Sigma \), \(\mu \in {\mathbb {R}}\). The optimal Hardy constant \(H:=(N-k-2)/2\) is deeply involved in the study of the Schrödinger operator \(L_\mu \). The Green kernel and Martin kernel of \(-L_\mu \) play an important role in the study of boundary value problems for nonhomogeneous linear equations involving \(-L_\mu \). If \(\mu \le H^2\) and the first eigenvalue of \(-L_\mu \) is positive then the existence of the Green kernel of \(-L_\mu \) is guaranteed by the existence of the associated heat kernel. In this paper, we construct the Martin kernel of \(-L_\mu \) and prove the Representation theory which ensures that any positive solution of the linear equation \(-L_\mu u = 0\) in \(\Omega {\setminus } \Sigma \) can be uniquely represented via this kernel. We also establish sharp, two-sided estimates for Green kernel and Martin kernel of \(-L_\mu \). We combine these results to derive the existence, uniqueness and a priori estimates of the solution to boundary value problems with measures for nonhomogeneous linear equations associated to \(-L_\mu \).
ER  -

GKIKAS, Konstantinos T. and Phuoc Tai NGUYEN. Martin kernel of Schrödinger operators with singular potentials and applications to B.V.P. for linear elliptic equations. \textit{Calculus of Variations and Partial Differential Equations}. Springer, 2022, vol.~61, No~1, p.~1-36. ISSN~0944-2669. Available from: https://dx.doi.org/10.1007/s00526-021-02102-6.

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