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

• Autores: Loredana Caso, Roberta D'Ambrosio, Maria Transirico
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 67, Fasc. 3, 2016, págs. 539-554
• Idioma: inglés
• DOI: 10.1007/s13348-015-0161-z
• Enlaces
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• Resumen

  • 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.

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