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

TL;DR

This paper constructs Green's functions for second order parabolic equations with singular lower order coefficients in unbounded domains, establishing Gaussian bounds under minimal regularity assumptions on the coefficients.

Contribution

It introduces a method to construct Green's functions for parabolic operators with critical Lebesgue space coefficients in unbounded domains, extending previous results.

Findings

Green's functions exist with Gaussian bounds in unbounded domains.

Lower order coefficients can be singular and belong to critical Lebesgue spaces.

Conditions on divergence of coefficients ensure the bounds.

Abstract

We construct Green's functions for second order parabolic operators of the form $Pu=\partial_t u-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c} \cdot \nabla u+du$ in $(-\infty, \infty) \times \Omega$, where $\Omega$ is an open connected set in $\mathbb{R}^n$. It is not necessary that $\Omega$ to be bounded and $\Omega = \mathbb{R}^n$ is not excluded. We assume that the leading coefficients $\bf A$ are bounded and measurable and the lower order coefficients $\boldsymbol{b}$, $\boldsymbol{c}$, and $d$ belong to critical mixed norm Lebesgue spaces and satisfy the conditions $d-{\rm div} \boldsymbol{b} \ge 0$ and ${\rm div}(\boldsymbol{b}-\boldsymbol{c}) \ge 0$. We show that the Green's function has the Gaussian bound in the entire $(-\infty, \infty) \times \Omega$.