Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms

TL;DR

This paper analyzes linear elliptic equations with general drifts and $L^1$-zero-order terms, establishing existence and uniqueness of solutions under various conditions, and providing explicit constants for error analysis.

Contribution

It introduces a transformation of general drifts into weak divergence-free drifts, enabling new existence and uniqueness results for elliptic equations with $L^p$ drifts and $L^1$-zero-order terms.

Findings

Existence and uniqueness of bounded weak solutions for divergence form equations.

Existence and uniqueness of strong solutions under VMO and differentiability conditions.

Explicit constants in estimates due to weak divergence-free property.

Abstract

This paper provides a detailed analysis of the Dirichlet boundary value problem for linear elliptic equations in divergence form with $L^p$-general drifts, where $p \in (d, \infty)$, and non-negative $L^1$-zero-order terms. Specifically, by transforming the general drifts into weak divergence-free drifts, we establish the existence and uniqueness of a bounded weak solution, showing that the zero-order term does not influence the quantity of the unique weak solution. Additionally, by imposing the $ VMO$ condition and mild differentiability on the diffusion coefficients and assuming an $L^s$-zero-order terms with $s \in (1, \infty)$, we demonstrate the existence and uniqueness of a strong solution for the corresponding non-divergence type equations. An important feature of this paper is that, due to the weak divergence-free property of the drifts in the transformed equations, the constants appearing in our estimates can be explicitly calculated, which is expected to offer significant applications in error analysis.