2021/06/02 by Pierluigi Colli, Colli, Pierluigi, T. Fukao +3 · 2 citations
Computer Science · Materials Science · Mathematics · #35B20 74N20 #35D30 #35K25 #35K61 #80A22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2106.01010
openalex publication_date 2021/06/02 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/01
An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn-Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward dynamic boundary condition at the limit. This is done in a very general setting, with nonlinear terms admitting maximal monotone graphs both in the bulk and on the boundary. The two graphs are related by a growth condition, with the boundary graph that dominates the other one. It turns out that in the limiting procedure the solution of the problem looses some regularity and the limit equation has to be properly interpreted in the sense of a subdifferential inclusion. However, the limit problem is still well-posed since a continuous dependence estimate can be proved. Moreover, in the case when the two graphs exhibit the same growth, it is shown that the solution enjoys more regularity and the boundary condition holds almost everywhere. An error estimate can also be shown, for a suitable order of the diffusion parameter.