Remarks on well-posedness for linear elliptic equations via divergence-free transformation

TL;DR

This paper explores the well-posedness of linear elliptic equations using a divergence-free transformation, overcoming limitations of classical methods especially for low integrability coefficients, and extends results via interpolation techniques.

Contribution

It introduces a divergence-free transformation approach that ensures well-posedness for elliptic equations with low integrability coefficients, extending classical results.

Findings

Divergence-free transformation establishes well-posedness for $c \,\in\, L^1(U)$.

Interpolation extends existence and uniqueness to $c \,\in\, L^s(U)$ for $s \in [1, \frac{2d}{d+2}]$.

Classical bilinear form methods face limitations with zero-order coefficients in $L^1(U)$.

Abstract

This paper investigates the well-posedness of linear elliptic equations, focusing on the divergence-free transformation introduced in the author's recent work [J. Math. Anal. Appl. 548 (2025), 129425]. By comparing this approach with classical bilinear form methods, we demonstrate that while standard techniques encounter limitations in handling zero-order coefficients $c \in L^1(U)$, the divergence-free transformation successfully establishes well-posedness in this setting. Furthermore, utilizing the Riesz-Thorin interpolation theorem between the cases $c \in L^1(U)$ and $c \in L^{\frac{2d}{d+2}}(U)$, we establish the existence and uniqueness of weak solutions under the assumption $c \in L^s(U)$ for $s \in [1, \frac{2d}{d+2}]$.