On the contraction properties for weak solutions to linear elliptic equations with $L^2$-drifts of negative divergence

TL;DR

This paper establishes the existence, uniqueness, and boundedness of weak solutions to certain linear elliptic equations with negative divergence $L^2$-drifts, demonstrating their $L^r$-contraction and $L^1$-stability properties.

Contribution

It introduces a novel approach using Dirichlet form theory to analyze contraction properties of solutions with negative divergence drifts.

Findings

Proves existence and uniqueness of solutions.

Establishes $L^r$-contraction properties.

Derives $L^1$-stability results.

Abstract

We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with $L^2$-drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the $L^r$-contraction properties of the unique weak solutions. Indeed, using the Dirichlet form theory, we construct a sub-Markovian $C_0$-resolvent of contractions and identify it to the weak solutions. Furthermore, we derive an $L^1$-stability result through an extended version of the $L^1$-contraction property.