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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 actingon $W^{0,\chi}_2(\Omega)$, called H\"{o}rmander space, have been the subject ofinvestigation by various authors. Then in this paper we will, under theassumption of parameter-ellipticity, establish results pertaining to theexistence and uniqueness of solutions of the boundary problem. Furthermore,under the further assumption that the boundary conditions are null, we willestablish results pertaining to the spectral properties of the Banach spaceoperator induced by the boundary problem, and in particular, to the angular andasymptotic distribution of its eigenvalues.

An elliptic boundary problem acting on generalized Sobolev spaces