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

In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form $$-\mathrm{div}(\gamma \mathrm{\nabla }u)+⟨\mathrm{\nabla }\phi +\mathbf{H},\mathrm{\nabla }u⟩+(c+\alpha )u=f\text{ in }U,u=0\text{on }\mathrm{\partial }U,$$ where $U\subset {\mathbb{R}}^d$ is bounded, $\gamma \in [1,\mathrm{\infty })$ is a constant, $\phi \in H^{1,2}(U)\cap L^{\mathrm{\infty }}(U)$, $\mathbf{H}\in L^p(U,{\mathbb{R}}^d)$ for some $p\in (d,\mathrm{\infty })$, 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 $\mathrm{\nabla }\phi$, 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.