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Some Flows in Shape Optimization

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Abstract

Geometric flows related to shape optimization problems of the Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele–Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain existence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed:we prove that the solutions converge to a generalized Bernoulli exterior free-boundary problem.

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Authors and Affiliations

  1. UFR des Sciences et Techniques, Université de Bretagne Occidentale, 6 Av. Le Gorgeu, BP 809, 29285, Brest, France

    Pierre Cardaliaguet

  2. Faculté des Sciences et Techniques, Université de Tours, Laboratoire de Mathématiques et Physique Théorique, Parc de Grandmont, 37200, Tours, France

    Olivier Ley

Authors
  1. Pierre Cardaliaguet
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  2. Olivier Ley
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Correspondence to Olivier Ley.

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Communicated by P-L. Lions

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Cardaliaguet, P., Ley, O. Some Flows in Shape Optimization. Arch Rational Mech Anal 183, 21–58 (2007). https://doi.org/10.1007/s00205-006-0002-z

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  • DOI: https://doi.org/10.1007/s00205-006-0002-z

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