Harnack Inequalities and Heat-kernel Estimates for Degenerate Diffusion Operators Arising in Population Biology

TL;DR

This paper establishes Harnack inequalities and heat kernel estimates for a class of degenerate elliptic operators on manifolds with corners, with applications in Population Biology, using advanced techniques in analysis.

Contribution

It extends existing methods to degenerate operators with non-vanishing transverse vector fields, providing new regularity and spectral results in complex geometric settings.

Findings

Weak solutions satisfy Harnack inequalities

Solutions belong to anisotropic Hölder spaces for positive times

Operators have a compact resolvent on $C^0$ and $L^2$ spaces

Abstract

This paper continues the analysis, started in [2, 3], of a class of degenerate elliptic operators defined on manifolds with corners, which arise in Population Biology. Using techniques pioneered by J. Moser, and extended and refined by L. Saloff-Coste, Grigoryan, and Sturm, we show that weak solutions to the parabolic problem defined by a sub-class of these operators, which consists of those that can be defined by Dirichlet forms and have non-vanishing transverse vector field, satisfy a Harnack inequality. This allows us to conclude that the solutions to these equations belong, for positive times, to the natural anisotropic Holder spaces, and also leads to upper and, in some cases, lower bounds for the heat kernels of these operators. These results imply that these operators have a compact resolvent when acting on $C^0$ or $L^2.$ The proof relies upon a scale invariant Poincare inequality that we establish for a large class of weighted Dirichlet forms, as well as estimates to handle certain mildly singular perturbation terms. The weights that we consider are neither Ahlfors regular, nor do they generally belong to the Muckenhaupt class $A_2.$