### Geometric Evolution Problems and Shape Optimization (GEPSO)

Project Type: Prin 2022

Funded by: MUR

Period: Sep 28, 2023 – Sep 27, 2025

Budget: €60.000,00

Principal Investigator: Nicola Fusco (Università degli Studi di Napoli Federico II)

Local coordinator: Matteo Novaga (Università di Pisa)

##### Participants

Giuseppe Buttazzo (Università di Pisa), Aldo Pratelli (Università di Pisa), Maria Stella Gelli (Università di Pisa), Massimo Gobbino (Università di Pisa)

##### Description

This project stems out from the collaboration of various people of the units that in recent years developed new ideas, techniques and methods in the study of geometric evolution problems, variational problems of isoperimetric type and shape optimization problems. The common background of all participants involved in this project led in a natural way to the formation of our research units and to the formulation of the present research program, divided in three main themes: geometric evolution problems and curvature flows of network, capillarity problems and isoperimetric inequalities, shape optimization. All these topics are strictly intertwined and share a large amount of techniques coming from Calculus of Variations, the Regularity theory of PDEs and Geometric Measure Theory. Moreover, apart from their intrinsic mathematical interest, some of the problems we intend to study originate from well known models developed in the fields of Materials Science and Structural Engineering.
1. GEOMETRIC EVOLUTION PROBLEMS AND CURVATURE FLOWS OF NETWORKS
Volume preserving mean curvature flow
Generalized nonlocal curvature flows
Crystalline curvature flow with stochastic forcing term
Mullins-Sekerka and Surface diffusion flow
Curvature flows of networks
2. CAPILLARITY AND ISOPERIMETRIC TYPE PROBLEMS
Capillarity type problems with a thin obstacle
Regularity of minimal configurations and validity of Young’s law
Isoperimetric type inequalities for the capillary energy
Nonlocal variants of the isoperimetric problem: thin films and minimization problems for attractive-repulsive nonlocal energies
3. SHAPE OPTIMIZATION
Two-phase shape optimization problems
Problems in geometric optimization and optimal domains in spectral analysis
Shape optimization problems with various boundary conditions