An elliptic boundary problem acting on generalized Sobolev spaces

TL;DR

This paper studies an elliptic boundary problem on generalized Sobolev spaces, establishing existence, uniqueness, and spectral properties of solutions under certain ellipticity and boundary conditions.

Contribution

It extends analysis of elliptic boundary problems to generalized Sobolev spaces with parameter-ellipticity, including spectral analysis of the associated operator.

Findings

Existence and uniqueness of solutions under parameter-ellipticity.

Spectral properties and eigenvalue distribution of the boundary problem operator.

Results for null boundary conditions on generalized Sobolev spaces.

Abstract

We consider an elliptic boundary problem over a bounded region $\Omega$ in $\mathbb{R}^n$ and acting on the generalized Sobolev space $W^{0,\chi}_p(\Omega)$ for $1 < p < \infty$. We note that similar problems for $\Omega$ either a bounded region in $\mathbb{R}^n$ or a closed manifold acting on $W^{0,\chi}_2(\Omega)$, called H\"{o}rmander space, have been the subject of investigation by various authors. Then in this paper we will, under the assumption of parameter-ellipticity, establish results pertaining to the existence and uniqueness of solutions of the boundary problem. Furthermore, under the further assumption that the boundary conditions are null, we will establish results pertaining to the spectral properties of the Banach space operator induced by the boundary problem, and in particular, to the angular and asymptotic distribution of its eigenvalues.