Abstract

We describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in the real N-dimensional space RN\mathbb{R}^N. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.

Author information

Contact details are reproduced from the original publication and may be historical.

Jimmy Lamboley

Dep. de Mathématiques, Ecole Normale Supérieure de Cachan, Campus de Ker-Lann, 35170 Bruz, France
and: IRMAR, Campus de Ker Lann, 35170 Bruz, France

jimmy.lamboley@bretagne.ens-cachan.fr

Michel Pierre

Dép. de Mathématiques, Ecole Normale Supérieure de Cachan, Campus de Ker-Lann, 35170 Bruz, France
and: IRMAR, Campus de Ker Lann, 35170 Bruz, France

michel.pierre@bretagne.ens-cachan.fr

Keywords

  • Shape optimization
  • shape derivatives
  • sets of finite perimeter
  • convex sets
  • Dirichlet energy

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Suggested citation

J. Lamboley, M. Pierre. “Structure of Shape Derivatives Around Irregular Domains and Applications.” Journal of Convex Analysis 14 (2007), No. 4, 807–822. https://doi.org/10.68381/jca14046

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press.