# Università degli studi di Pavia

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## Curriculum Analisi Matematica

**Brief description of the areas of research in Mathematical Analysis of the members from Milano Bicocca and Como**

The research activities of the members of the College of Como and Milano Bicocca in the framework of the Mathematical Analysis and its applications can, in short, be described as follows.

At Milano Bicocca there is an active research group in Partial Differential Equations that deals in particular with elliptic equations and conservation laws. The group members present in the College Rinaldo M. Colombo, Veronica Felli, Mauro Garavello e Graziano Guerra.

As for the equations of elliptic type, one is interested in the development of variational methods for the study of problems of existence, qualitative properties of solutions and spectral properties (spectral stability with respect to perturbations of the domain or operator) for equations or systems of partial differential equations of elliptic type (such as Schrödinger equations, linear and non-linear).

As for the conservation laws, problems motivated by vehicular traffic (even on networks), crowd dynamics and gas-dynamic, lead to analytical and numerical ivestigation of equations and systems of conservation laws. Of current interest are the well-posedness, the qualitative properties of solutions and control problems.

At Milan-Bicocca there is an active research group in harmonic analysis and applications, consisting of: Leonardo Colzani (harmonic analysis in Euclidean spaces, developments in eigenfunctions, convergence of series and Fourier integral), Bianca Di Blasio (harmonic analysis on Lie groups, in particular nilpotent and solvable groups), Luigi Fontana (analysis on CR spheres, in particular precise estimates involving Sobolev spaces in these areas, for example, Adams-Moser-Trudinger type inequlities) and Stefano Meda (harmonic analysis on manifolds and functional calculus for Laplace-Beltrami operators).

Various aspects of global analysis and differential geometry and the study of differential equations on manifolds (linear and non-linear, often related to problems of geometric nature, such as the problem of Yamabe) are the focus of the research activity of Alberto Setti (Como).

The convex analysis in sub-Riemannian structures (for example, the Heisenberg group, or, more generally, the Carnot groups) is one of the areas within which the activity of Rita Pini (Milano Bicocca) takes place.

**Brief description of the areas of research in mathematical analysis of the members from Pavia**

Formed with the help of some great Italian mathematicians (Enrico Magenes, Claudio Baiocchi, Franco Brezzi), the "School of Pavia" in the field of Partial Differential Equations is now made up of a very active research group, both in the theoretical analysis and in the modeling and applied developments, in collaboration with numerous internationally renowned experts.

The more theoretical aspects (addressed with varying degrees by all the researchers of the group) are generally

- well-posedness, and the qualitative properties (regularity, singular limits and homogenization, asymptotic behavior, optimal control problems) of the solutions to boundary value problems, typically nonlinear, both evolutionary (parabolic and hyperbolic) and stationary (elasticity, free boundary and free discontinuity problems)

- the structure of the models, their formulation and their derivation (variational inequalities, phase-field, gradient flows, energy and viscosity formulations of rate-independent problems).

This type of investigation involves

- sharp methods of real and functional analysis, convex optimization and calculus of variations, and the theory of regularity, which are developed also independently from more specific models (Gamma-convergence, homogenization, singular perturbations, contraction semigroups, functional inequalities, Dirichlet forms, convergence and approximation estimates, convex analysis, analysis in metric spaces).

The main applications concern in particular

- phase transition models, fluid mechanics, damage and delamination phenomena, liquid crystals, pattern formation, reaction diffusion systems (Elena Bonetti, Pierluigi Colli, Gianni Gilardi, Giulio Schimperna, Antonio Segatti, Marco Veneroni),

- Problems rate-independent, in particular phenomena of elastic-plasticity, damage and fractures, problems of discontinuity free (Maria Giovanna Mora, Matteo Negri, Giuseppe Savaré, Enrico Vitali)

- Parabolic problems of the type of porous media, fast diffusion, p-Laplacian, drift-diffusion, Fokker-Planck (Simona Fornaro, Ugo Gianazza)

- optimal transport theory, analysis and geometry of metric spaces of measures (Fabio Cavalletti, Stefano Lisini, Giuseppe Savaré)