# Masarykova univerzita

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

Autoři GKIKAS Konstantinos T. 2022 Článek v odborném periodiku Calculus of Variations and Partial Differential Equations https://link.springer.com/article/10.1007/s00526-021-02102-6 http://dx.doi.org/10.1007/s00526-021-02102-6 Schrödinger operators; Singular elliptic equations; Green's functions; Boundary value problems for second-order elliptic equations 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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