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A Boundary Control Problem for the Equation and Dynamic Boundary Condition of Cahn–Hilliard Type

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Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

Part of the book series: Springer INdAM Series ((SINDAMS,volume 22))

Abstract

A dynamic boundary condition is a type of partial differential equation that describes the dynamics of a system on the boundary. Combining with the heat equation in a smooth-bounded domain, the characteristic structure of “total mass conservation” appears, namely, the volume in the bulk plus the volume on the boundary is conserved. Based on this interesting structure, an equation and dynamic boundary condition of Cahn–Hilliard type was introduced by Goldstein–Miranville–Schimperna. In this paper, based on the previous work of Colli–Gilardi–Sprekels, a boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type is considered. The optimal boundary control that realizes the minimal cost under a control constraint is determined, and a necessary optimality condition is obtained.

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Acknowledgements

This paper is dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. The authors are grateful to the anonymous referee for reviewing the original manuscript and for valuable comments that helped to clarify and refine this paper. This work is supported by JSPS KAKENHI Grants-in-Aid for Scientific Research(C), Grant Number 26400164 for TF and 26400179 for NY.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto, 612-8522, Japan

    Takeshi Fukao

  2. Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686, Japan

    Noriaki Yamazaki

Authors
  1. Takeshi Fukao
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  2. Noriaki Yamazaki
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Corresponding author

Correspondence to Takeshi Fukao .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

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Fukao, T., Yamazaki, N. (2017). A Boundary Control Problem for the Equation and Dynamic Boundary Condition of Cahn–Hilliard Type. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_10

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