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Shape Differentiability of Lagrangians and Application to Overdetermined Problems

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Mathematical Analysis of Continuum Mechanics and Industrial Applications III (CoMFoS 2018)

Part of the book series: Mathematics for Industry ((MFI,volume 34))

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Abstract

A class of geometry-dependent Lagrangians is investigated in a functional analysis framework with respect to the property of shape differentiability. General results are presented due to Delfour–Zolésio who adopted to shape optimization an abstract theorem of Correa–Seeger on the directional differentiability. A crucial point concerns the bijective property of function spaces as well as their feasible sets that must be preserved under a kinematic flow of geometry. The shape differentiability result is applied to overdetermined free-boundary and inverse problems expressed by least-square solutions. The theory is supported by explicit formulas obtained for calculation of the shape derivative.

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Acknowledgments

V.A.K. is supported by the Austrian Science Fund (FWF) project P26147-N26: “Object identification problems: numerical analysis” (PION) and the Austrian Academy of Sciences (OeAW). K.O. is supported by the JSPS KAKENHI Grant Number 16K05285. The joint work began in CoMFoS18 that is the workshop by the Activity group MACM (Mathematical Aspects of Continuum Mechanics) of JSIAM. The authors thank the Japan Society for the Promotion of Science (JSPS) research project (No. J19-721) joint with the Russian Foundation for Basic Research (RFBR) project (N 19-51-50004)

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Authors and Affiliations

  1. Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstr. 36, Graz, 8010, Austria

    Victor A. Kovtunenko

  2. Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Victor A. Kovtunenko

  3. Faculty of Information Design and Sociology, Hiroshima Kokusai Gakuin University, 6-20-1, Aki-ku, Hiroshima, 739-0321, Japan

    Kohji Ohtsuka

Authors
  1. Victor A. Kovtunenko
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  2. Kohji Ohtsuka
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Corresponding author

Correspondence to Victor A. Kovtunenko .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Tokyo University of Science, Tokyo, Japan

    Hiromichi Itou

  2. College of Science and Engineering, Ritsumeikan University, Kusatsu, Japan

    Shiro Hirano

  3. Faculty of Mathematics and Physics, Kanazawa University, Kanazawa, Ishikawa, Japan

    Masato Kimura

  4. NAWI Graz, Institut für Mathematik, Karl-Franzens University of Graz, Graz, Austria

    Victor A. Kovtunenko

  5. Siberian Branch of RAS, Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia

    Alexandr M. Khludnev

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Kovtunenko, V.A., Ohtsuka, K. (2020). Shape Differentiability of Lagrangians and Application to Overdetermined Problems. In: Itou, H., Hirano, S., Kimura, M., Kovtunenko, V.A., Khludnev, A.M. (eds) Mathematical Analysis of Continuum Mechanics and Industrial Applications III. CoMFoS 2018. Mathematics for Industry, vol 34. Springer, Singapore. https://doi.org/10.1007/978-981-15-6062-0_7

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