Existence and uniqueness of solutions of degenerate elliptic equations with lower order terms

TL;DR

This paper establishes the existence and uniqueness of solutions for a class of degenerate elliptic equations with lower order terms under weak assumptions, extending previous results in the literature.

Contribution

It generalizes existing results by weakening the hypotheses on coefficients and Sobolev inequalities for degenerate elliptic equations.

Findings

Proves existence and uniqueness under weak hypotheses.

Shows weaker Sobolev inequalities suffice with stronger coefficient integrability.

Extends the theory of degenerate elliptic equations to broader conditions.

Abstract

We prove the existence and uniqueness of solutions to a Dirichlet problem \[ \begin{cases} Lu = f + v^{-1}\text{Div}(v{\bf e} h), & x \in \Omega; u = 0, & x \in \partial \Omega, \end{cases}\] where $L$ is a degenerate, linear, second order elliptic operator with lower order terms. We assume very weak hypotheses, in terms of the coefficients of the equation, and we also assume the existence of degenerate Sobolev and Poincar\'e inequalities. One notable feature of our result is that we show that we can assume significantly weaker versions of the Sobolev inequality if we in turn assume stronger integrability conditions on the coefficients. Our theorems generalize a number of results in the literature on degenerate elliptic equations.