11 jun 2018 - 21 jun 2018
DiMaI Università di Firenze
Schedule: Mo-Fr 10:30-12:30
Place: aula Tricerri DiMaI, V.le Morgagni 67A - Firenze
Program: In these lectures, isoperimetric inequalities will be seen from a shape optimisation point of view. After an introduction of the main tools, I will present recent results (based on free boundary techniques) which can be used to obtain qualitative information on the optimal domain minimising a shape functional: existence of a solution, (mild) regularity and numerical approximation, with a particular focus on the analysis of shape sub and supersolutions. As main examples, we shall discuss problems issued from spectral geometry, like the minimisation of the k-th eigenvalue of the Laplace operator with Dirichlet boundary conditions, under a volume constraint. I will also show how existence and regularity of an optimal shape for a (very) particular class of functionals, implies that the shape is a ball! This argument works for the first eigenvalue of the Dirichlet Laplacian, and gives a proof of the Faber-Krahn inequality which does not use rearrangements. As well, I will discuss isoperimetric inequalities of this type for the Robin-Laplacian. Shape optimisation problems with Robin boundary conditions require a different approach, based on free discontinuity techniques. Using the argument described above for the Faber-Krahn inequality, I will prove that the ball minimises the first Robin eigenvalue among domains of prescribed measure. In particular I will detail a monotonicity formula which is the key point for the (Ahlfors) regularity of the optimal set. I will also show how this techniques have a crucial impact on extracting quantitative forms for those isoperimetric inequalities. Depending on time, I will also discuss multiphase shape optimization problems andor shape optimization in the class of convex sets. This kind of questions rise different difficulties, related to collective behaviours of shapes or a good understanding of optimality conditions.
Speakers: Dorin Bucur.