Shape Optimization (I. Fragalà, F. Tomarelli)

Shape optimization is a branch of mathematical analysis which has obtained a great impulse in the last two decades:  it consists in minimizing some functional defined on a class of admissible shapes, namely subsets of the Euclidean space. The functional usually comes from some physical model, and may bring into play some geometric features of the shape (such as volume or surface area), as well as some PDE’s defined on it (classical examples are the electrostatic capacity, or the eigenvalues of the Laplacian with Dirichlet or Neumann conditions). In many cases, the involved PDE’s are settled on domains whose boundary is partially unknown, or which contains an unknown interface of physical relevance, for instance separating two distinct phases. Such unknown sets are usually referred to as ``free boundaries’’: their study occupies an important role in modern analysis, due to the richness of both its theoretical and applied aspects. 

Strictly related to shape optimization, a field in rapid development is that of geometric-functional inequalities. Indeed, on one hand such inequalities often provide sharp estimates for various geometric quantities of arbitrary shapes; on the other hand, their study  has allowed to improve various techniques of frequent use in shape optimization, such as those based on perturbations of domains and related variational formulas.

Thorough connections must be pointed out with different contact fields, such as  Convex Geometry (that branch of Mathematics which studies convex bodies, i.e. compact convex sets in the Euclidean space), and Numerical Analysis (since recently some rigorous proofs of functional inequalities have  been carried over with the help of numerical implementations).

Some of the topics under study which can be framed in this  research line are:

  • reverse-type isoperimetric inequality for eigenvalues;
  • inequalities involving the notion of polarity for convex bodies;
  • optimal partition problems.
Politecnico di Milano - Dipartimento di Matematica