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(A center dot del u+ bu)-c center dot del u-du, inanopen set Omega subset of R-n, n >= 3, with possibly infinite Lebesgue measure. We assume that the n x n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b, c is an element of L-loc(n,infinity) (Omega) and d is an element of L-loc(n/2,infinity) (Omega), or vertical bar b vertical bar(2), vertical bar c vertical bar(2), vertical bar d vertical bar is an element of kappa(loc)(Omega), where kappa(loc)(Omega) stands for the local Stummel-Kato class. Let kappa(Dini)(Omega) be a variant of kappa(Omega) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of Lu = f - divg, where f and vertical bar g vertical bar(2) is an element of kappa(Dini)(Omega) if, for q is an element of [n, infinity), any of the following assumptions holds: (i) vertical bar b vertical bar(2), vertical bar d vertical bar is an element of kappa(Dini)(Omega) and either c is an element of L-loc(n,q) (Omega) or vertical bar c vertical bar(2) is an element of kappa(loc)(Omega); (ii) divb + d <= 0 and either b + c is an element of L-loc(n,q) (Omega) or vertical bar b + c vertical bar(2) is an element of kappa(loc)(Omega); (iii) -divc + d <= 0 and vertical bar b + c vertical bar(2) is an element of kappa(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 -divc + d <= 0, we construct the Green's function associated with L satisfying quantitative estimates. Under the additional hypothesis vertical bar b + c vertical bar(2) is an element of kappa' (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 associated bilinear form.