A Priori Bounds in and in for Solutions of Elliptic Equations

Sara Monsurrò,Maria Transirico

doi:10.1155/2013/650870

Abstract

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

Highlights

The aim of this work is to give an overview on some recent results dealing with the study of a certain kind of the Dirichlet problem in the framework of unbounded domains
Let us define the moduli of continuity of functions belonging to Mq,λ(Ω) or M∘q,λ(Ω)
If Ω is an open subset of Rn having the cone property and g ∈ Mr(Ω), with r > p if p = n, the operator in (17) is bounded from W1,p(Ω) to Lp(Ω)

Summary

Introduction

The aim of this work is to give an overview on some recent results dealing with the study of a certain kind of the Dirichlet problem in the framework of unbounded domains. In two recent works, [4, 5], considering a more regular set Ω and supposing that the lower order terms coefficients are as in [3] for n ≥ 3 and as in [2] for n = 2, we prove that if f ∈ L2(Ω) ∩ L∞(Ω), there exists a constant C, whose dependence is completely described, such that. We provide two applications of our final Lp-bound, p > 1, recalling the results of [7, 8] where our estimate plays a fundamental role in the study of certain weighted and non-weighted non-variational problems with leading coefficients satisfying hypotheses of Miranda’s type (see [9]). In [13,14,15] a very general weighted case, with principal coefficients having vanishing mean oscillation, has been taken into account

A Class of Spaces of Morrey Type

The Variational Problem

Non-Variational Problems