Remarks on well-posedness for linear elliptic equations via divergence-free transformation
Haesung Lee
TL;DR
This work analyzes the well-posedness of linear elliptic Dirichlet problems of the form $-\mathrm{div}(A \nabla u) + \langle H, \nabla u \rangle + c u = f$ in a bounded domain $U$, focusing on allowing the zero-order coefficient $c$ to lie in the low-regularity space $L^1(U)$. It compares classical bilinear-form methods with a divergence-free transformation that converts the original problem into an equivalent one with better-behaved coefficients, enabling existence and uniqueness results even when $c$ is only in $L^1(U)$; the approach is extended to $c$ in the interpolated range $L^s(U)$ for $s \in [1, \tfrac{2d}{d+2}]$ via Riesz–Thorin interpolation. The paper also develops a structural condition-based divergence-free transformation that yields transformed problems with positive weight $\rho$ and a drift term $\mathbf{B}$, establishing well-posedness and a priori estimates in a broader setting than the classical theory. An interpolation theorem further blends the endpoint regimes $c \in L^{\tfrac{2d}{d+2}}(U)$ and $c \in L^1(U)$ to obtain existence, uniqueness, and bounds for intermediate $c$ with explicit exponents. Overall, the results substantially relax zero-order integrability requirements while preserving solvability and estimates, offering new tools for elliptic problems with drift terms.
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 $s \in [1, \frac{2d}{d+2}]$.