Embedding Theorems and Boundary-value Problems for cusp domains

TL;DR

This paper investigates Robin boundary-value problems on domains with isolated singularities, providing precise conditions on boundary weights for existence and uniqueness, thereby advancing the understanding of elliptic problems in cusp domains.

Contribution

It offers an exact characterization of boundary weights for domains with peaks, enabling proper formulation of Robin problems in singular domains.

Findings

Characterization of weights for boundary trace spaces.

Conditions for existence and uniqueness of Robin problems.

Application to elliptic operators in cusp domains.

Abstract

We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, \xi}(\partial D)$ existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $I_1: H^{1}(D)\to L^{2}(D)$ and $I_{2}:H^{1}(D)\to L^{2,\xi}(\partial D)$ i.e. on type of singularities. We obtain an exact description of the weights $\xi$ for bounded domains with 'outside peaks' on its boundaries. This result allows us to formulate correctly the corresponding Robin boundary-value problems for elliptic operators.