Green functions for the heat and Laplace equations with dynamical boundary conditions in a ball
Xuzhou Yang
TL;DR
This work develops a framework for Green's functions and related kernels for Laplace and heat equations with dynamical boundary conditions on the unit ball $B_1$. It first constructs an explicit Laplace dynamical Green's function $K_1(x,y,t)$ and the Dirichlet heat kernel $\Gamma_1$, then formulates two approximation schemes for the heat Green's function $G_1$ and the solution $u$ by combining $\Gamma_1$, the Dirichlet heat kernel, and boundary-coupling kernels. An eigenfunction expansion for $G_1$ is presented via a Neumann-type eigenproblem on $\partial B_1$, providing a principled representation in terms of $\psi_k$ and $\lambda_k$. The approximations $\mathcal{G}_1$ and $\tilde{\Gamma}_1,\tilde{H}_1$ offer practical tools for analyzing and implementing solutions to the dynamical-boundary heat problem in a ball, with probabilistic and analytic perspectives interwoven throughout. The results lay groundwork for probabilistic interpretations and potential extensions to nonlinear or more complex geometries.
Abstract
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.