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

We consider elliptic operators in divergence form with lower order terms of the form $Lu=-$div$\mathrm{\nabla }u+bu)-c\mathrm{\nabla }u-du$, 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$$\mathcal{K}${loc}(\Omega)$standsforthelocalStummel-Katoclass\mathrm{.}Let$$\mathcal{K}${Dini}(\Omega)$beavariantof$$\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 $\mathrm{\mid }f\mathrm{\mid }$ and $|g|^2\in $\mathcal{K}${Dini}(\Omega)$if,for$q\in [n, \infty)$, any of the following assumptions holds: a) $\mathrm{\mid }b{\mathrm{\mid }}^2,\mathrm{\mid }d\mathrm{\mid }\in {\mathcal{K}}_{Dini}(\mathrm{\Omega })$ and either $c\in L_{loc}^{n,q}(\mathrm{\Omega })$ or $\mathrm{\mid }c{\mathrm{\mid }}^2\in {\mathcal{K}}_{loc}(\mathrm{\Omega })$; b) div$b +d \leq 0$andeither$b+c\in L^{n,q}{loc}(\Omega)$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$c+d\le 0$, we construct the Green’s function associated with $L$ satisfying quantitative estimates. Under the additional hypothesis $\mathrm{\mid }b+c{\mathrm{\mid }}^2\in {\mathcal{K}}^{\mathrm{\prime }}(\mathrm{\Omega })$, 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 $\mathrm{\Omega }$, while we do not assume smallness of the norms of the coefficients or coercivity of the bilinear form.