Well-posedness of linear elliptic equations with $L^d$-drifts under divergence-type conditions

TL;DR

This paper proves the well-posedness of linear elliptic equations with critical $L^d$-drifts and low-regularity zero-order coefficients using divergence-free transformations and structural drift conditions.

Contribution

It introduces a novel approach combining divergence-free transformations and structural drift conditions to establish well-posedness under minimal regularity assumptions.

Findings

Existence and uniqueness of weak solutions are established.

The method relaxes regularity assumptions on zero-order coefficients.

A new weight function construction via the weak maximum principle is developed.

Abstract

We establish the well-posedness of linear elliptic equations with critical-order drifts in $L^d$ and positive zero-order coefficients in $L^1$ or $L^{\frac{2d}{d+2}}$, where classical methods are often too restrictive. Our approach relies on a divergence-free transformation and a structural condition on the drift vector field, which admits a decomposition into a regular component and another whose weak divergence belongs to $L^{\tilde{q}}$ for some $\tilde{q} > \frac{d}{2}$. This condition is essential for constructing a suitable weight function $\rho$ via the weak maximum principle and the Harnack inequality. Within this framework, we prove the existence and uniqueness of weak solutions, significantly relaxing the regularity assumptions on the zero-order coefficients in $L^{\frac{d}{2}}$.