Well-posedness in weighted Sobolev spaces for elliptic equations of Cordes type
Tóm tắt
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.
Tài liệu tham khảo
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