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}} $.
Citation: Haesung Lee. Well-posedness of linear elliptic equations with $ L^d $-drifts under divergence-type conditions[J]. Electronic Research Archive, 2025, 33(12): 7974-7998. doi: 10.3934/era.2025351
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}} $.
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Haesung Lee. Well-posedness of linear elliptic equations with $ L^d $-drifts under divergence-type conditions[J]. Electronic Research Archive, 2025, 33(12): 7974-7998. doi: 10.3934/era.2025351
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