Compactness of the embedding operators for rough domains (Q2711485)

New classes of non-smooth bounded domains \(D\), for which the embedding operator from \(H^1(D)\) into \(L^2(D)\) is compact, are introduced. These classes include, in particular, the domains whose boundary locally are graphs of continuous functions, but also contain much larger classes of domains, for example, in the two-dimensional case, the bounded domains whose boundaries are locally graphs of piecewise-continuous functions with ''jump''-type discontinuity at a finite number of points. Examples of non-smooth domains for which the above embedding is compact are given. Applications to the existence and uniqueness of the solutions to the scattering problem in the exterior of rough obstacles are given and a larger class of rough obstacles in scattering theory than it was done earlier is also considered.