Weighted p--Laplace approximation of linear and quasi-linear elliptic problems with measure data

TL;DR

This paper introduces a weighted p-Laplace approximation method for solving linear and quasi-linear elliptic problems with measure data, ensuring convergence to weak or entropy solutions depending on the data's integrability.

Contribution

It proposes a novel approximation technique using weighted p-Laplace terms and a specific diffeomorphism to handle measure data in elliptic problems.

Findings

The approximation converges to a weak solution for general measure data.

The method converges to an entropy solution when the data is in L1.

The approach effectively handles heterogeneous anisotropic diffusion matrices.

Abstract

We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted p--Laplace term. A well chosen diffeomorphism between R and (--1, 1) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in L 1 .