Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes

TL;DR

This paper establishes well-posedness of elliptic boundary value problems with Uhlenbeck structure on convex polytopes in weighted Sobolev spaces and introduces a convergent finite element method for their nonlinear variants.

Contribution

It proves well-posedness in weighted Sobolev spaces for elliptic problems with singular forcing and develops a convergent finite element discretization for nonlinear cases.

Findings

Well-posedness of linear and nonlinear elliptic problems in weighted Sobolev spaces.

Convergence of the proposed finite element discretization.

Analysis of discretization in weighted spaces for linear problems.

Abstract

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$ with $p \in (1,\infty$). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.