Embedding Theorems and Boundary-value Problems for 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.

Correction of misprints: pages 5, 6, 11. We found more compact of Theorem 1, part 3, 4