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

Mihalis Mourgoglou

Published 2019-04-09Version 1

We consider elliptic operators in divergence form with lower order terms of the form $Lu = -\text{div} (A \cdot \nabla u + b u ) - c\cdot \nabla u - du$, 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, $b, c \in L^n(\Omega)$, $d \in L^\frac{n}{2}(\Omega)$, and $ \mathcal{K}_{Dini,1/2}(\Omega)$ stands for a variant of the Stummel-Kato class with a Dini-type condition. If $\text{div} b +d \leq 0$ holds, or if $-\text{div} c + d \leq 0$ and $|b+c|^2 \in \mathcal{K}_{Dini,1/2}(\Omega)$ hold, then we develop a De Giorgi/Nash/Moser theory for solutions of $Lu = f - \text{div} g$, where $ f$ and $|g|^2 \in \mathcal{K}_{Dini,1/2}(\Omega)$. The most important feature of our proofs is that all the estimates are scale invariant and independent of $\Omega$. We also show that the variational Dirichlet problem is well-posed and, for its solution, we prove a Wiener-type criterion for boundary regularity. Finally, assuming that $|b+c|^2 \in \mathcal{K}_{Dini,1/2}(\Omega)$, we construct Green's functions associated with operators of the form above and show, among others, global pointwise bounds.