The Fundamental Solution of an Elliptic Equation with Singular Drift

TL;DR

This paper investigates the existence and behavior of fundamental solutions for elliptic equations with singular drift terms, extending classical results to operators with less regular coefficients and mild singularities.

Contribution

It introduces a new singular integral criterion for the existence of fundamental solutions in elliptic operators with singular drift and provides examples illustrating the perturbative effects of the drift.

Findings

Singular integral controls fundamental solution existence.

Singular drift can be a mild perturbation or enable fundamental solutions.

Examples demonstrate the drift's impact on fundamental solutions.

Abstract

For $n\geq 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, ${\mathcal L}(x,\partial_x)=a_{ij}(x)\partial_i\partial_j+b_k(x)\partial_k$, where the $a_{ij}$ have modulus of continuity $\omega(r)$ satisfying the square-Dini condition and the $b_k$ are allowed mild singularities of order $r^{-1}\omega(r)$. A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift $b_k\partial_k$ may act as a perturbation that does not dramatically change the fundamental solution of ${\mathcal L}^o=a_{ij}\partial_i\partial_j$, or it could change an operator ${\mathcal L}^o$ that does not have a fundamental solution to one that does.