# Shape optimization under convexity constraint

## Jimmy Lamboley (Université Paris-Dauphine)

created by depascal on 21 Mar 2009

21 mar 2009

Abstract.

Seminari di Calcolo delle Variazioni

Mercoledi' 25 Marzo, Sala riunioni Dipartimento di Matematica

Dr. Jimmy Lamboley (ENS Cachan, antenne de Bretagne)

Title : Shape optimization under convexity constraint

Abstract : Shape optimization is the study of optimization problems whose unknown is a domain of \Rd. I will focus on the case where admissibles shapes are required to be convex set of $\R^2$. Under this constraint, it is hard to write optimality conditions. In a first part, I will show how we can write such conditions (first and second order), and I will use these ones to exhibit a class of functionals which leads to polygonal optimal shapes (work with A. Novruzi). In a second part, I will focus on the minimization of the second eigenvalue for the Laplace operator (Dirichlet conditions), model problem which show difficulties linked to convexity constraint, and also difficulties due to the regularity of optimal shapes. We particularly show that optimal shapes are C{1,12} and no more, for this problem. I will end with some links with partially overdetermined problems (work with I. Fragalà and F. Gazzola).

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