Embedding theorems and boundary-value problems for cusp domains

We study the Robin boundary-value problem for bounded domains with isolated singularities. Because trace spaces of space (D)on boundaries of such domains are weighted Sobolev spaces L²ξ (∂D), existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators I₁ : W¹₂) → L ²(D)and I₂ : W¹₂ (D) → L ²'ξ (∂D) i.e. on types of singularities. We obtain an exact description of weights ξ 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 I₁,I₂, we prove also that these Robin boundary-value problems with the spectral parameter are of Fredholm type.