Green's function for second order elliptic equations with singular lower order coefficients and applications

Abstract: We will discuss Green's function for second order elliptic operators of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$ in domains $\Omega\subseteq\mathbb R^n$, for $n\geq 3$. We will assume that $A$ is elliptic and bounded, and also that $d\geq\text{div}b$ or $d\geq\text{div}c$ in the sense of distributions.

In the setting of Lorentz spaces, we will explain why the assumption $b-c\in L^{n,1}(\Omega)$ is optimal in order to obtain a pointwise bound of the form $G(x,y)\leq C|x-y|^{2-n}$. Under the assumption $d\geq\text{div}b$, we will also discuss why this assumption is necessary to even have weak type bounds on Green's function. Finally, for the case $d\geq\text{div}c$, we will deduce a maximum principle and a Moser type estimate, showing again that the assumption $b-c\in L^{n,1}(\Omega)$ is optimal.

Our estimates will be scale invariant and no regularity on $\partial\Omega$ will be imposed. In addition, $\mathcal{L}$ will not be assumed to be coercive, and there will be no smallness assumption on the lower order coefficients.

analysis of PDEsclassical analysis and ODEs

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.