Publication details

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


GKIKAS Konstantinos T. NGUYEN Phuoc Tai

Year of publication 2022
Type Article in Periodical
Magazine / Source Calculus of Variations and Partial Differential Equations
MU Faculty or unit

Faculty of Science

Keywords Schrödinger operators; Singular elliptic equations; Green's functions; Boundary value problems for second-order elliptic equations
Description 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 \(-L_\mu \).
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