# Partial differential equations and calculus of variations

Differential equations and related optimization problems are fundamental in the description of various phenomena in very different fields of application. The research team aims to study equations and variational problems related to models that appear naturally in Materials Sciences, Biology, Chemistry, Biomedical and Finance, without neglecting the structural and qualitative aspects of more purely theoretical variational problems and historical/didactic/informative aspects involving the foundations of these disciplines. The web page mathematical-analysis.unimore.it contains the updated information on the research activity of our team. The group of researchers is highly skilled in the following specific research topics:

Special materials and regularity in the Calculus of Variations

- Regularity for minimizers of integrals functional with general growth and for solutions to obstacle problems
- Asymptotic analysis, dimensional reduction and homogenization for problems arising in material sciences

Optimization problems in calculus of variations

- Shape optimization for functionals involving geometric quantities such as perimeter, inradius, area of a set in a suitable class of domains
- Study of functional spaces with BMO type seminorms: characterization of Sobolev spaces with this kind of seminorms and applications to image processing
- Study of free boundary problems with Robin boundary conditions on the free interface: existence and regularity of minimizers

Mathematical models of Financial Markets and regularity theory

- Degenerate Kolmogorov equations and stochastic processes
- Obstacle problems
- Black & Scholes model
- Numerical solutions to Kolmogorov equations
- Skewness and risk measurement

Nonlinear evolution equations and applications

- Analysis of solutions to the Nonlinear Schroedinger Equation with singular, double-well, periodic potentials
- Existence and uniqueness for systems of Kohn-Sham type equations
- Nonlinear phase-transition models, in particular thermodynamically consistent systems with diffuse interface
- Porous media models with thermomechanical interactions and phase-transition
- Existence and controllability of semilinear equations in Banach spaces
- Tumor growth models

Regularity theory for partial differential equations

- Degenerate Kolmogorov operators
- p-Laplace operators
- Obstacle problem

Elliptic PDEs

- Elliptic PDEs:
- Variational formulation, qualitative properties, existence, classification, non-existence, multiplicity for equations and systems of nonlinear elliptic PDEs involving the Laplacian, p-Laplacian, fractional and higher order Laplacian operators
- Elliptic problems of Sobolev-critical and exponential growth, elliptic problems with loss of compactness, functional spaces, sharp inequalities and existence/non-existence of extremal functions
- Asymptotic properties of the spectrum of the Laplacian with Dirichlet boundary conditions on bounded domains

**Staff researchers:**

Carlo Benassi, Michela Eleuteri, Stefania Gatti, Serena Guarino Lo Bianco, Luca La Rocca, Maria Manfredini, Carlo Mercuri, Stefania Perrotta, Sergio Polidoro, Andrea Sacchetti, Federica Sani, Massimo Villarini.

[Ultimo aggiornamento: 04/04/2023 11:20:57]