Boundary estimates for a degenerate parabolic equation with partial Dirichlet boundary conditions

TL;DR

This paper investigates boundary regularity and estimates for weak solutions of degenerate parabolic equations with partial Dirichlet conditions, relevant to population genetics models of gene frequency evolution.

Contribution

It provides new boundary regularity results and a priori estimates for solutions of degenerate parabolic equations with partial boundary conditions, applicable to genetic models.

Findings

Derived boundary regularity properties for solutions.

Established a priori pointwise supremum estimates.

Applied results to describe transition probabilities and hitting distributions.

Abstract

We study the boundary regularity properties and derive a priori pointwise supremum estimates of weak solutions and their derivatives in terms of suitable weighted $L^2$-norms for a class of degenerate parabolic equations that satisfy homogeneous Dirichlet boundary conditions on certain portions of the boundary. Such equations arise in population genetics in the study of models for the evolution of gene frequencies. Among the applications of our results is the description of the structure of the transition probabilities and of the hitting distributions of the underlying gene frequencies process, which correspond to the fundamental solution and the caloric measure of the parabolic equation, respectively.