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

Article type: Research Article

Authors: Giga, Yoshikazu^a | Łasica, Michał^{a; b} | Rybka, Piotr^{c; *}

Affiliations: [a] Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan | [b] Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland | [c] Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Correspondence: [*] Corresponding author. E-mail: [email protected].

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 Γ-convergence of the functionals generating these flows. Our analysis of strong solutions depends on a new version of the Reilly identity.

Keywords: Boundary layer, dynamic boundary conditions, convergence of gradient flows, Reilly identity, Γ-convergence

DOI: 10.3233/ASY-231862

Journal: Asymptotic Analysis, vol. 135, no. 3-4, pp. 463-508, 2023