NO^3 - Nodal Optimization, NOnlinear elliptic equations, NOnlocal geometric problems, with a focus on regularity

Funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 del 2/2/2022

We are interested in several problems of analysis of PDEs and Calculus of Variations arising in physics and other sciences, pertaining to the following macro-areas:

• A. Asymptotic behavior of solutions of competing systems, analysis of nodal sets, and spectral properties of singularly perturbed problems. This includes the regularity of interfaces in phase-separation models and optimal partition problems, the study of nodal sets for solutions to elliptic problems, and the qualitative and quantitative analysis of spectral properties under singular perturbation of the reference domain.
• B. Shape optimization and free-boundary problems, with a focus on shape optimization problems coming from electrodynamics and quantum models with aggregation/scattering competition, and on biological models with various type of diffusion.
• C. Theoretical aspect of nonlinear elliptic problems, with a focus on three prototypical problems: existence and properties of critical points for nonlinear Schrödinger equations on various type of domains, or even on networks; blow-up phenomena or compactness of solutions for Yamabe-type equations; regularity theory for the mean curvature equation in Lorentz-Minkowski space.
• D. Nonlocal geometric problems, with a focus on the regularity theory for nonlocal minimal surfaces, the classification of entire solutions of the fractional Allen-Cahn equation, and the analysis of the nonlocal mean curvature flow, both from the point of view of the regularity, and from the point of view of the long-time behavior.

In each point, we wish to understand the theoretical aspects, and their possible impact on the model. A main feature of this project stays in the fact that, despite the variety of problems we plan to attack, there is a remarkable methodological unity. The different issues all require mastery in variational methods and critical points theory, regularity and qualitative theory for elliptic and parabolic equations, blow-up analysis and monotonicity formulas, geometric measure theory