An Lp‐Estimate for Weak Solutions of Elliptic Equations

Sara Monsurrò,Maria Transirico

doi:10.1155/2012/376179

Abstract

We prove an Lp‐a priori bound, p > 2, for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

Highlights

The Dirichlet problem for second order linear elliptic partial differential equations in divergence form and with discontinuous coefficients in bounded open subsets of Rn, n ≥ 2, is a classical problem that has been widely studied by several authors we refer, e.g., to 1–6 .In this paper, we want to analyze certain aspects of the same kind of problem, but in the framework of unbounded domains.More precisely, given an unbounded open subset Ω of Rn, n ≥ 2, we are interested in the study of the elliptic second order linear differential operator in variational form L n − i,j ∂ ∂xj aij∂ ∂xi dj i n 1 bi∂ ∂xi c, 1.1 with coefficients aij ∈ L∞ Ω, and in the following associated Dirichlet problem u ∈ ◦
We extend a known result by Stampacchia see 1, or 13 for details, obtained within the framework of the generalization of the study of certain elliptic equations in divergence form with discontinuous coefficients on a bounded open subset of Rn to some problems arising for harmonic or subharmonic functions in the theory of potential. This is done in order to obtain a preliminary lemma, proved in Section 3, that permits to consider some particular test functions in the variational formulation of our problem
2.5 In Lemma 2.2 below, we show a further generalization of 2.4, always in the case of unbounded domains

Summary

Introduction

The Dirichlet problem for second order linear elliptic partial differential equations in divergence form and with discontinuous coefficients in bounded open subsets of Rn, n ≥ 2, is a classical problem that has been widely studied by several authors we refer, e.g., to 1–6. We want to analyze certain aspects of the same kind of problem, but in the framework of unbounded domains. Given an unbounded open subset Ω of Rn, n ≥ 2, we are interested in the study of the elliptic second order linear differential operator in variational form. ∂ ∂xi c, 1.1 with coefficients aij ∈ L∞ Ω , and in the following associated Dirichlet problem u.

Abstract and Applied Analysis

Simple calculations give uh

We also associate to L the bilinear form

Fs u dx