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

This paper continues the analysis, started in [3, 4], 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, Grigor’yan, 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 nonvanishing 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 Hölder 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 C0 or L. The proof relies upon a scale invariant Poincaré 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 A2. ∗Research partially supported by NSF grant DMS12-05851, and ARO grant W911NF-12-1-0552. Address: Department of Mathematics, University of Pennsylvania; e-mail: [email protected] †Research partially supported by NSF grant DMS1105050. Address: Department of Mathematics, Stanford University; e-mail: [email protected]