Weighted a priori bounds for elliptic operators of Cordes type

Loredana Caso,Roberta D’Ambrosio,Maria Transirico

doi:10.1186/s13660-015-0758-5

Loredana Caso, Roberta D’Ambrosio + Show 1 more

Open Access

https://doi.org/10.1186/s13660-015-0758-5

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Abstract

We prove some weighted $W^{2,2}$ -a priori bounds for a class of elliptic second order linear differential operators of Cordes type on unbounded domains of $\mathbb {R}^{n}$ , $n\geq2$ .

Highlights

Consider the Dirichlet problem u ∈ W (Ω ) ∩ ◦, (Ω), Lu = f, f ∈ L (Ω), ( )where Ω is an open subset of Rn, n ≥, with a suitable regularity property, and L is an elliptic second order linear differential operator, with measurable coefficients, defined by n ∂
Where Ω is an open subset of Rn, n ≥, with a suitable regularity property, and L is an elliptic second order linear differential operator, with measurable coefficients, defined by n
Our aim in this paper is to prove some weighted a priori estimates for a problem similar to ( ) on unbounded domains of Rn, n ≥, where the leading coefficients are in the class of discontinuity of Cordes type

Summary

Introduction

Where Ω is an open subset of Rn, n ≥ , with a suitable regularity property, and L is an elliptic second order linear differential operator, with measurable coefficients, defined by n. The author studied the solvability of problem ( ) in the planar case, only assuming the following condition on the leading coefficients: aij = aji ∈ L∞(Ω), i, j = , , together with the boundedness of the lower order terms ai, a of operator L defined in ( ) (see [ ]). If Ω is an unbounded domain, problem ( ) was studied, for instance, in [ ], where the leading coefficients satisfy Cordes type condition and the lower order terms ai, a belong to suitable classes of Morrey type spaces. Our aim in this paper is to prove some weighted a priori estimates for a problem similar to ( ) on unbounded domains of Rn, n ≥ , where the leading coefficients are in the class of discontinuity of Cordes type. Taking into account the results of this paper, we are in a position to approach the study of solvability of problem ( )

Function spaces

Weight functions and weighted Sobolev spaces