Well-posedness in weighted Sobolev spaces for elliptic equations of Cordes type

Abstract

In this paper we prove some weighted \(W^{2,2}\)-a priori bounds for a class of linear, elliptic, second-order, differential operators of Cordes type in certain weighted Sobolev spaces on unbounded open sets \(\varOmega \) of \(\mathbb {R}^{n},\,n\ge 2\). More precisely, we assume that the leading coefficients of our differential operator satisfy the so-called Cordes type condition, which corresponds to uniform ellipticity if \(n=2\) and implies it if \(n\ge 3\), while the lower order terms are in specific Morrey type spaces. Here, our analytic technique mainly makes use of the existence of a topological isomorphism from our weighted Sobolev space, denoted by \(W^{2,2}_s(\varOmega )\) (\(s\in \small \mathbb {R}\)), whose weight is a suitable function of class \(C^2(\bar{\varOmega })\), to the classical Sobolev space \(W^{2,2}(\varOmega )\), which allow us to exploit some well-known unweighted a priori estimates. Using the above mentioned \(W^{2,2}_s\)-a priori bounds, we also deduce some existence and uniqueness results for the related Dirichlet problems in the weighted framework.