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.
H.Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland/Elsevier, Amsterdam–London/New York, 1973.
2.
P.Colli and J.-F.Rodrigues, Diffusion through thin layers with high specific heat, Asymptotic Anal.3(3) (1990), 249–263. doi:10.3233/ASY-1990-3304.
3.
D.Fujiwara and H.Morimoto, An -theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math.24(3) (1977), 685–700.
4.
P.Grisvard, Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées 22, Masson, Paris; Springer-Verlag, Berlin, 1992.
5.
M.Ljulj, E.Marušić-Paloka, I.Pažanin and J.Tambača, Mathematical model of heat transfer through a conductive pipe, ESAIM Math. Model. Numer. Anal.55(2) (2021), 627–658. doi:10.1051/m2an/2021008.
6.
L.Matthias, Passing from bulk to bulk-surface evolution in the Allen–Cahn equation, NoDEA Nonlinear Differential Equations Appl.20(3) (2013), 919–942. doi:10.1007/s00030-012-0189-7.
7.
A.Mielke, On evolutionary Γ-convergence for gradient systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lect. Notes Appl. Math. Mech., Vol. 3, Springer, Cham, 2016, pp. 187–249. doi:10.1007/978-3-319-26883-5_3.
8.
R.C.Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J.26(3) (1977), 459–472. doi:10.1512/iumj.1977.26.26036.
9.
E.Sandier and S.Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg–Landau, Comm. Pure Appl. Math.57(12) (2004), 1627–1672. doi:10.1002/cpa.20046.
10.
G.Savaré and A.Visintin, Variational convergence of nonlinear diffusion equations: Applications to concentrated capacity problems with change of phase, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.8(1) (1997), 49–89.
11.
S.Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst.31(4) (2011), 1427–1451. doi:10.3934/dcds.2011.31.1427.
12.
R.E.Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, Vol. 49, AMS, Providence, RI, 1997.
