Grant Details

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