Sharp kernel bounds for parabolic operators with first order degeneracy

TL;DR

This paper establishes precise bounds for the fundamental solutions of a class of degenerate parabolic operators with first-order terms, extending understanding of their behavior under boundary conditions.

Contribution

It provides sharp upper and lower estimates for the parabolic kernel of a singular elliptic operator with first-order degeneracy in a half-space setting.

Findings

Derived sharp bounds for the parabolic kernel.

Extended estimates to operators with boundary conditions.

Enhanced understanding of degenerate parabolic operators.

Abstract

We prove sharp upper and lower estimates for the parabolic kernel of the singular elliptic operator \begin{align*} \mathcal L&=\mbox{Tr }\left(AD^2\right)+\frac{\left(v,\nabla\right)}y, \end{align*} in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$ under Neumann or oblique derivative boundary conditions at $y=0$.