Composition Operators on Sobolev Spaces and Neumann Eigenvalues

Abstract

We study spectral properties of the divergence form elliptic operators −div[A(z)∇f (z)] with the Neumann boundary condition in (non)convex domains Ω ⊂ C. The suggested method is based on the composition operators on Sobolev spaces with applications to the Poincaré inequalities. 0

On the Neumann $(p,q)$-eigenvalue problem in Hölder singular domains

“…The weak quasiconformal mappings permits us to "transfer" the Sobolev-Poincaré inequalities from one domain to another. In the work [17], the authors obtained the following result. Theorem 3.2.…”

“…Let us formulate the following result from the work [17] in the refined form because of the very useful remark by Charles Fefferman. Theorem 3.3.…”

Composition Operators on Sobolev Spaces And

“…and belongs to the Sobolev space L 1 p ′ ( Ω) [15]. Because the inverse mapping is a mapping of finite distortion, then by [19]…”

Space quasiconformal composition operators with applications to Neumann eigenvalues

“…Applications are based on the geometric theory of composition operators on Sobolev spaces [20,34,38] in the special case of operators generated by quasiconformal mappings. Composition operators on Sobolev spaces permit us to give estimates of norms for embedding operators of Sobolev spaces into Lebesgue spaces in a large class of space domains that includes domains with Hölder singularities [22,23]. Quasiconformal mappings allow us to describe the important subclass of these embedding domains in the terms of quasihyperbolic geometry.…”