Embedding theorems and boundary-value problems for cusp domains

Abstract:

We study the Robin boundary-value problem for bounded domains with isolated singularities. Because trace spaces of space $W_{2}^{1}(D)$ on boundaries of such domains are weighted Sobolev spaces $L^{2,\xi }(\partial D)$, existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $I_{1}:W_{2}^{1}(D)\rightarrow L^{2}(D)$ and $I_{2}:W_{2}^{1}(D)\rightarrow L^{2,\xi }(\partial D)$ i.e. on types of singularities. We obtain an exact description of 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. Using compactness of embedding operators $I_{1},I_{2}$, we prove also that these Robin boundary-value problems with the spectral parameter are of Fredholm type.