Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains

Abstract

We consider elliptic operators in divergence form with lower order terms of the form $Lu=-$div$\nabla u+bu)-c\nabla u-du$, in an open set $\Omega\subset \mathbb{R}^n$, $n\geq 3$, with possibly infinite Lebesgue measure. We assume that the $n\times n$ matrix $A$ is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either $b,c\in L^{n,\infty}_{loc}(\Omega)$ and $d\in L_{loc}^{\frac{n}{2},\infty}(\Omega)$, or $|b|^2,|c|^2,|d|\in \mathcal{K}_{loc}(\Omega)$, where $\mathcal{K}_{loc}(\Omega)$ stands for the local Stummel-Kato class. Let $\mathcal{K}_{Dini}(\Omega)$ be a variant of $\mathcal{K}(\Omega)$ satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of $Lu=f-$div$g$, where $|f|$ and $|g|^2\in \mathcal{K}_{Dini}(\Omega)$ if, for $q\in [n, \infty)$, any of the following assumptions holds: a) $|b|^2,|d|\in \mathcal{K}_{Dini}(\Omega)$ and either $c\in L^{n,q}_{loc}(\Omega)$ or $|c|^2\in \mathcal{K}_{loc}(\Omega)$; b) div$b +d \leq 0$ and either $b+c\in L^{n,q}_{loc}(\Omega)$ or $|b+c|^2\in \mathcal{K}_{loc}(\Omega)$; c) $-$div$c+d \leq 0$ and $|b+c|^2\in \mathcal{K}_{Dini}(\Omega)$. We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming $-$div$c+d\leq 0$, we construct the Green's function associated with $L$ satisfying quantitative estimates. Under the additional hypothesis $|b+c|^2\in \mathcal{K}'(\Omega)$, we show that it satisfies global pointwise bounds and also construct the Green's function associated with the formal adjoint operator of $L$. An important feature of our results is that all the estimates are scale invariant and independent of $\Omega$, while we do not assume smallness of the norms of the coefficients or coercivity of the bilinear form.