Stability, regularity and symmetry issues in geometric variational problems

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This project is aimed at investigating stability, regularity, and symmetry issues in various geometric variational problems, and at exploiting the corresponding results in the effective description of equilibrium configurations of surface tension driven physical systems. For example, the stability theory for isoperimetric-type problems recently established by the PI and his collaborators, beyond its intrinsic mathematical interest, has revealed useful in studying minimizers of classical sharp interface energies, of the Gates-Lebowitz-Penrose energy, the Ohta-Kawasaki energy and variants, and in cavitation models in Non-linear Elasticity. This research program will achieve significant improvements in the stability theory for geometric inequalities, broadening the reach of the theory to include new and challenging situations, and opening new spaces for further applications to problems of applied interest. Specific stability issues considered in this project arise in the study of minimizing clusters, Plateau's problem, and isoperimetric problems in arbitrary codimension and in Gauss space. The project will also advance the mathematical theory of capillarity problems, by addressing regularity issues related to the validity of Young's law, and by providing a quantitative description of geometric properties of equilibrium configurations. Finally, the project aims to some conclusive developments in symmetrization theory, by characterizing, from a geometric viewpoint, those situations where equality cases in symmetrization inequalities imply symmetry of minimizers. This project aims to advance the mathematical understanding of geometric variational problems. Geometric variational problems play a fundamental role in the mathematical modeling of Nature, and in particular, in our quantitative and qualitative understanding of equilibrium states of physical systems. Despite their ubiquitous interest, and the very considerable amount of work that has been devoted to their study both from mathematicians, physicists, and engineers, several questions remain unanswered, or just partially understood, due to the mathematical challenges they arise. In turn, geometric variational problems play also a pivotal role in various area of Mathematics, including Analysis, Probability Theory, and Geometry. Several important contributions to the stability theory for geometric variational problems has been obtained in recent years by the PI and his collaborators, with applications to the effective description of equilibrium states of physical systems, and with the introduction of new mathematical ideas and techniques. An important part of this project will consist in the training of graduate students on these new mathematical developments.