Shape derivatives for minima of integral functionals

Ilaria Fragalà (1) , Ilaria Lucardesi (2, 3) , Guy Bouchitté (3)

1 POLIMI - Politecnico di Milano [Milan]
2 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies
3 IMATH - Institut de Mathématiques de Toulon - EA 2134

For Ω varying among open bounded sets in R n , we consider shape functionals J(Ω) defined as the infimum over a Sobolev space of an integral energy of the kind Ω [f (∇u) + g(u)], under Dirichlet or Neumann conditions on ∂Ω. Under fairly weak assumptions on the integrands f and g, we prove that, when a given domain Ω is deformed into a one-parameter family of domains Ω ε through an initial velocity field V ∈ W 1,∞ (R n , R n), the corresponding shape derivative of J at Ω in the direction of V exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of V on ∂Ω. Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.

Mots clés: duality; domain derivative; infimum problems; shape functionals

Domaines: Analyse classique [math.CA]; Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Mécanique des structures [physics.class-ph]

Ilaria Fragalà, Ilaria Lucardesi, Guy Bouchitté. Shape derivatives for minima of integral functionals. Mathematical Programming, 2014, pp.27. ⟨10.1007/s10107-013-0712-6⟩. ⟨hal-01127679⟩