Notes on linear elliptic equations with $L^2$-gradient perturbations and singular zero-order coefficients

Abstract

In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form \[ -\operatorname{div}(γ\nabla u)+\langle \nablaϕ+\mathbf{H},\nabla u\rangle+(c+α)u=f \quad\text{ in }U, \quad\; u=0 \quad\text{on }\partial U, \] where $U\subset {\mathbb{R}}^d$ is bounded, $\gamma \in [1,\infty )$ is a constant, $\varphi \in H^{1,2}(U)\cap L^{\infty }(U)$, $\mathbf{H}\in L^p(U,{\mathbb{R}}^d)$ for some $p\in (d,\infty )$, and $c\in L^1(U)$ with $c\ge 0$. A key feature of this setting is that the drift contains the low-regularity term $\nabla \varphi$, which is only assumed to belong to $L^2(U,{\mathbb{R}}^d)$, while the zero-order coefficient is merely integrable. We prove that, even under these rough assumptions, well-posedness and quantitative energy and $L^2$ estimates remain valid. In addition, by using a previously established interpolation result, we characterize a trade-off between the integrability of the source term $f$ and that of the zero-order coefficient $c$, and show that well-posedness together with the corresponding quantitative estimates hold under these interpolated assumptions.

11 pages