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.