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

TL;DR

This paper constructs Green's functions for second order elliptic equations with singular lower order coefficients, providing pointwise and Lorentz space bounds, and extends results to gradient bounds under Dini conditions.

Contribution

It introduces a method to construct Green's functions with singular lower order coefficients and establishes bounds under minimal regularity assumptions.

Findings

Constructed Green's function with pointwise bounds

Established Lorentz space bounds for Green's function

Derived gradient bounds under Dini regularity conditions

Abstract

We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients $\mathbf A$ is uniformly elliptic and bounded and the lower order coefficients $\boldsymbol{b}$, $\boldsymbol{c}$, and $d$ belong to certain Lebesgue classes and satisfy the condition $d - \nabla \cdot \boldsymbol{b} \ge 0$. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green's function in the case when the mean oscillations of the coefficients $\mathbf A$ and $\boldsymbol{b}$ satisfy the Dini conditions and the domain is $C^{1, \rm{Dini}}$.