Searcharxiv⌕ Search

arXiv subjects

Xuzhou Yang

Publications and source records attributed to Xuzhou Yang.

3 recordsLinked to original sources
arXiv2025

Green functions for the heat and Laplace equations with dynamical boundary conditions in a ball

The green functions for the heat and Laplace equations with dynamical boundary conditions in a ball are studied. First, the green functions of the Laplace equation with a dynamical boundary condition are given, and the properties of related heat kernels are discussed. Then using these ingredients, two complementary approximations to the heat equation with a dynamical boundary condition in a ball are constructed, including an approximation of green function and an approximation of solution. Moreover, the green function of the heat equation with a dynamical boundary condition is implicitly presented by a series of eigenfunctions.

math.AP↗
arXiv2024

Schauder estimate for the boundary diffusion equaiton

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.

math.AP↗