On the energy of a flow arising in shape optimization

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

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

In Cardaliaguet-Ley (2006) we have defined a viscosity solution for the gradient flow of the exterior Bernoulli free boundary problem. We prove here that the associated energy is non decreasing along the flow. This justifies the "gradient flow" approach for such kind of problem. The proof relies on the construction of a discrete gradient flow in the flavour of Almgren-Taylor-Wang (1993) and on proving it converges to the viscosity solution.

Mots clés: non local geometric evolution; front propagation; gradient flow; minimizing movement; capacity potential; viscosity solutions; shape optimization; free boundary problem; Bernoulli problem

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Pierre Cardaliaguet, Olivier Ley. On the energy of a flow arising in shape optimization. Interfaces Free Bound., 2008, 10 (2), pp.223-243. ⟨hal-00084972⟩