Existence, uniqueness, regularity and stability of solutions to linear $X$-elliptic equations with measurable coefficients

TL;DR

This paper establishes existence, uniqueness, regularity, and stability results for solutions to linear $X$-elliptic equations with measurable coefficients, highlighting how solution regularity improves with data summability.

Contribution

It provides the first comprehensive analysis of solutions to linear $X$-elliptic equations with $L^1$ data, including stability and regularity results.

Findings

Solutions depend continuously on the data.

Improved data summability leads to better solution regularity.

Results extend classical elliptic theory to $X$-elliptic equations.

Abstract

We prove an existence and uniqueness result for solutions to linear $X$-elliptic equations with $L^1$ data and zero Dirichlet boundary conditions. Such solutions depend continuously on the datum. Moreover, we show that an improvement in the summability of the data yields a corresponding improvement in the summability of the solutions, in a manner analogous to the one that occurs in the case of uniformly elliptic equations.