# H\"older regularity for the linearized porous medium equation in bounded domains

## TL;DR

This paper establishes H"older regularity for weak solutions of a linearized porous medium equation in bounded domains, addressing singular and degenerate cases with boundary conditions.

## Contribution

It provides the first systematic proof of H"older regularity for solutions of linearized porous medium equations with boundary conditions.

## Key findings

- Weak solutions are H"older continuous in bounded domains.
- Regularity results apply to both divergence and nondivergence forms.
- Addresses singular and degenerate cases of the equation.

## Abstract

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized porous medium equation with the homogeneous Dirichlet boundary condition. We prove the H\"older regularity of their weak solutions.