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We consider elliptic operators in divergence form with lower order terms ofthe 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 assumethat the $n\times n$ matrix $A$ is uniformly elliptic with real, merely boundedand possibly non-symmetric coefficients, and either $b,c\inL^{n,\infty}_{loc}(\Omega)$ 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 aCarleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory forsolutions of $Lu=f-$div$g$, where $|f|$ and $|g|^2\in\mathcal{K}_{Dini}(\Omega)$ if, for $q\in [n, \infty)$, any of the followingassumptions 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 boundaryregularity. Assuming global conditions on the coefficients, we show that thevariational Dirichlet problem is well-posed and, assuming $-$div$c+d\leq 0$, weconstruct the Green's function associated with $L$ satisfying quantitativeestimates. Under the additional hypothesis $|b+c|^2\in \mathcal{K}'(\Omega)$,we show that it satisfies global pointwise bounds and also construct theGreen's function associated with the formal adjoint operator of $L$. Animportant feature of our results is that all the estimates are scale invariantand independent of $\Omega$, while we do not assume smallness of the norms ofthe coefficients or coercivity of the bilinear form.

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