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@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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