H\"older regularity of solutions of degenerate parabolic equations of general dimension

TL;DR

This paper proves regularity and estimate results for a class of degenerate parabolic equations in multiple dimensions, extending classical theory to equations with non-divergence form and degenerate structure.

Contribution

It establishes key estimates like Alexandroff-Bakelman-Pucci, Harnack inequality, and Hölder regularity for degenerate parabolic equations of non-divergence form in all dimensions.

Findings

Proved Hölder regularity of solutions.

Established Harnack inequality for the class.

Derived Schauder estimates for solutions.

Abstract

We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, the H\"older regularity and the Schauder estimates to a class of degenerate parabolic equations of non-divergence form in all dimensions \begin{equation} \mathcal{L}u:= u_t -Lu= u_t -(x a_{11} u_{xx} +2\sqrt{x} \sum_{j=2}^n a_{1j} u_{x y_j} + \sum_{i,j=2}^n a_{ij} u_{y_i y_j} + b_1 u_x +\sum_{j=2}^n b_j u_{y_j} ) =g\ \end{equation} on \(x \geq 0, y=(y_2,\ldots, y_n) \in \mathbb{R}^{n-1}\), with bounded measurable coefficients.