A compact embedding theorem for generalized Sobolev spaces

Abstract

We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative quadratic forms on R n . Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegener- ate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains in R n , the latter for weights equal to powers of the distance to the boundary. We also derive a compactness result for Lebesgue spaces on quasimetric spaces unrelated to R n and possibly without any notion of gradient. The Rellich-Kondrachov theorem is frequently used to study the existence of solutions to elliptic equations, a famous example being subcritical and critical Yamabe equations, resulting in the solution of Yamabe's problem; see (Y), (T), (A), (S). Further applications lie in proving the existence of weak solutions to Dirichlet problems for elliptic equations with rough boundary data and coefficients; see (GT). In a sequel to this paper, we will apply our compact embedding results to study the existence of solutions for some classes of degenerate equations.