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 ) W_{2}^{1}(D) on boundaries of such domains are weighted Sobolev spaces L 2 , ξ ( ∂ D ) 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 ) → L 2 ( D ) I_{1}:W_{2}^{1}(D)\rightarrow L^{2}(D) and I 2 : W 2 1 ( D ) → L 2 , ξ ( ∂ D ) 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 ξ \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 I_{1},I_{2} , we prove also that these Robin boundary-value problems with the spectral parameter are of Fredholm type.