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

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Abstract

We establish the well-posedness of linear elliptic equations with critical-order drifts in L^d and positive zero-order coefficients in L¹ or $L^{\frac{2 d}{d+2}}$, 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 ρ 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 $L^{\frac{d}{2}}$.

Keywords

linear elliptic equations / well-posedness / weak solutions / divergence-free transformation / Harnack inequality / weak maximum principle

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