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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F22%3A00119376" target="_blank" >RIV/00216224:14310/22:00119376 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/article/10.1007/s00526-021-02102-6" target="_blank" >https://link.springer.com/article/10.1007/s00526-021-02102-6</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00526-021-02102-6" target="_blank" >10.1007/s00526-021-02102-6</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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 (0le k &lt; 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 ).

Název v anglickém jazyce

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

Popis výsledku anglicky

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 (0le k &lt; 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 ).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-14413Y" target="_blank" >GJ19-14413Y: Lineární a nelineární eliptické rovnice se singulárními daty a související problémy</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Calculus of Variations and Partial Differential Equations

ISSN

0944-2669

e-ISSN

1432-0835

Svazek periodika

61

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

36

Strana od-do

1-36

Kód UT WoS článku

000717551300005

EID výsledku v databázi Scopus

2-s2.0-85119156947