Some duality perspectives for non convex problems in calculus of

Atelier
« Nouveaux défis pour le calcul des variations provenant de problèmes en science des matériaux et
traitement de l’image — En l’honneur du e anniversaire d’Irene Fonseca »
— mai 
Workshop
“New Challenges for the Calculus of Variations Stemming From Problems in the Materials Sciences and
Image Processing — In Honour of the th Birthday of Irene Fonseca”
May —, 
Some duality perspectives for non convex problems
in calculus of variations
Guy Bouchitté*
[email protected]
In this talk I will present a duality theory for problems of the kind
𝐽(Ω, 𝜇) ∶= inf 􏿼􏾙 􏿴𝑓(∇𝑢) + 𝑔(𝑢)􏿷 𝑑𝑥 − 􏾙 𝑢 𝑑𝜇 , 𝑢 = 0 on 𝜕Ω�
􏸵
􏸵
where 𝑔, 𝛾 are possibly non convex functions with suitable growth conditions and 𝑓 is a
convex integrand on ℝ𝑑 . Our aim is to characterize global minimizers of such a problem
and study the stability of the minimal value with respect to small variations of the domain
Ω or of the source term 𝜇.
Our duality scheme is formulated in Ω × ℝ on which the dual problem consists in maximizing a transport flow under suitable convex constraints. Applications with numerical
issues will be given for a class of free boundary problems.
This is joint work with Ilaria Fragala (Politecnico di Milano- Italy) and Minh Phan (Toulon).
*
IMATH, Université du Sud Toulon-Var, BP 20132 - 83957, Bâtiment U, La Garde, 83957, FRANCE.