Schauder estimate for the boundary diffusion equaiton

TL;DR

This paper establishes Schauder estimates for a linear boundary diffusion equation, crucial for analyzing nonlinear boundary diffusion problems like boundary heat control and Yamabe flow, enabling proofs of short-term existence of smooth solutions.

Contribution

It introduces $C^{1+eta}$-type Schauder estimates for boundary diffusion equations, facilitating analysis of nonlinear problems and proving short-time existence of smooth solutions.

Findings

Schauder estimates derived for boundary diffusion equations.

Application to prove short-time existence of solutions.

Extension to nonlinear boundary diffusion problems.

Abstract

In this paper, a certain type of linear boundary diffusion equation is studied. Such equation is crucial in the research of a non-linear boundary diffusion problem which was originated from the boundary heat control problem and Yamabe flow. Such equation is also analogous to the classical fast diffusion equation and porous medium equation. Under an compatible energy condition and the help of two types of test functions, $C^{1+\alpha}$-type Schauder estimates of the solution to the linear boundary diffusion equation in local configuration are obtained, through iteration to integrals both on the interior and the boundary at the same time. As applications, Schauder estimates of a new $C^{1+\alpha}$-type solution to the related boundary diffusion equations are established, which eventually provide a proof of short time existence of this new type of solution with smooth positive initial data and inhomogeneous boundary term.