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Shape Derivative of the Dirichlet Energy for a Transmission Problem

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Abstract

For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed in terms of u. The main difficulties in this context arise from the weak regularity of the domain and the possibly non-empty intersection of the graph of u and the transmission interface. The explicit formula for the shape derivative is then used to identify the partial differential equation solved by the minimizers of an energy functional arising in the modeling of micromechanical systems.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, 31062, Toulouse Cedex 9, France

    Philippe Laurençot

  2. Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Christoph Walker

Authors
  1. Philippe Laurençot
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  2. Christoph Walker
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Correspondence to Philippe Laurençot.

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Communicated by G. Dal Maso.

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Partially supported by the CNRS Projet International de Coopération Scientifique PICS07710.

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Laurençot, P., Walker, C. Shape Derivative of the Dirichlet Energy for a Transmission Problem. Arch Rational Mech Anal 237, 447–496 (2020). https://doi.org/10.1007/s00205-020-01512-8

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