Some flows in shape optimization

Pierre Cardaliaguet (1) , Olivier Ley (2, 3)

1 LM - Laboratoire de mathématiques de Brest
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 IRMAR - Institut de Recherche Mathématique de Rennes

Geometric flows related to shape optimization problems of 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.

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Cardaliaguet, Olivier Ley. Some flows in shape optimization. Archive for Rational Mechanics and Analysis, 2007, 183 (1), pp.21-58. ⟨10.1007/s00205-006-0002-z⟩. ⟨hal-00456121⟩