The heat equation with the dynamic boundary condition as a singular limit of problems degenerating at the boundary

TL;DR

This paper investigates how dynamic boundary conditions for the heat equation can be derived as limits of boundary layer problems, analyzing convergence and functional limits with a new Reilly identity.

Contribution

It introduces a novel approach to derive dynamic boundary conditions as a singular limit, including convergence analysis and a new version of the Reilly identity.

Findings

Convergence of weak and strong solutions as boundary layer width tends to zero

Establishment of $ ext{Γ}$-convergence of associated functionals

Development of a new Reilly identity for analyzing strong solutions

Abstract

We derive the dynamic boundary condition for the heat equation as a limit of boundary layer problems. We study convergence of their weak and strong solutions as the width of the layer tends to zero. We also discuss $\Gamma$-convergence of the functionals generating these flows. Our analysis of strong solutions depends on a new version of the Reilly identity.