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. a Phuoc Tai NGUYEN

Základní údaje

Originální název

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

Autoři

GKIKAS, Konstantinos T. (300 Řecko, garant) a Phuoc Tai NGUYEN (704 Vietnam, domácí)

Vydání

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

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Německo

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 2.100

Kód RIV

RIV/00216224:14310/22:00119376

Organizační jednotka

Přírodovědecká fakulta

DOI

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

UT WoS

000717551300005

Klíčová slova anglicky

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

Štítky

rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 29. 11. 2021 09:49, Mgr. Marie Šípková, DiS.

Anotace

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 \(-L_\mu \).

Návaznosti

GJ19-14413Y, projekt VaV

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

Investor: Grantová agentura ČR, Lineární a nelineární eliptické rovnice se singulárními daty a související problémy

Citovat

GKIKAS, Konstantinos T. a 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, roč. 61, č. 1, s. 1-36. ISSN 0944-2669. Dostupné z: 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. a 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, roč.~61, č.~1, s.~1-36. ISSN~0944-2669. Dostupné z: https://dx.doi.org/10.1007/s00526-021-02102-6.

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