Pisa-Freiburg
school in Applied Analysis

Giuseppe Buttazzo

Title: Optimization problems for elliptic PDEs

Abstract: The goal is to present optimization problems related to elliptic PDEs with Dirichlet boundary conditions. More precisely, the goal is to consider the problems:

L1. PDE $-div(a(x)∇u)=f where we look for the optimal coefficient a in suitable classes;

L2. PDE -∆u+V(x)u=f where we look for the optimal potential V in suitable classes;

L3. PDE -∆u=f where we look for the optimal right-hand side f in suitable classes;

L4. PDE -∆u=f where we look for the optimal domain Ω in suitable classes.

Existence of optimal solutions will be discussed, together with some properties of them. In the last lecture some young researchers will present some results on related questions.

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L4 will consist of three talks by Ilaria Lucardesi, Davide Carazzato, Roberto Ognibene

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Davide Carazzato: "Characterization of the maximizers for a Wasserstein energy"

Abstract. We study an energy defined through an optimal transport problem, which previously appeared in some models representing biological membranes. We prove that the maximizers coincide with a ball in many relevant cases, and this is done by means of a symmetrization technique. This talk is based on a joint work with Almut Burchard and Ihsan Topaloglu.

Ilaria Lucardesi: "Shape optimization under width constraint"

Abstract. Among the geometric constraints considered in shape optimization, the classical ones are surely the perimeter and the volume. For example, the literature on minimizing the eigenvalues of the Laplacian under one of these constraints spans more than a century of Mathematics. Less common, but equally significant, are the constraints of maximal width (the diameter) and minimal width, which describe how 'large' and 'small' an admissible set can be, respectively. The shape optimization of monotone functionals (with respect to inclusion) under maximal/minimal width constraint naturally leads to the consideration of two classes of shapes: the bodies of constant width and the reduced bodies. The classical literature on shape optimization in these familes of sets focuses on functionals of purely geometric type. The coupling with functionals of spectral type is a new research field and has many open questions. In this seminar, I will present some recent results and ongoing projects in this direction. The talk is based on joint works with Antoine Henrot (Nancy, France) and Davide Zucco (Turin, Italy).

Roberto Ognibene: "On the stability of Robin eigenvalues"

Abstract. In this talk, I will discuss the asymptotic behavior of the eigenvalues of the Laplacian with Robin boundary conditions, with respect to the Robin parameter. In particular, I will focus on two main cases: when the parameter goes to zero (convergence to Neumann eigenvalues) or to infinity (convergence to Dirichlet eigenvalues).