Equazioni Nonlineari alle Derivate Parziali

Research topics

This group is mainly concerned with the study of nonlinear P.D.E.s issued from mathematical models of physical phenomena; this often requires an interplay between analytical and physical properties.

Free boundary problems.

These are boundary value problems for differential equations, which are set in a domain whose boundary is a priori unknown, and is accordingly named a free boundary an additional quantitative condition is then provided to exclude nondeterminacy.

Problems of this sort arise in a large number of phenomena of applicative interest. Examples include the classical Stefan problem and more general models of phase transitions: here the free boundary is represented by the moving interface between phases. Another important example is provided by filtration through porous media, where free boundaries occur as fronts between saturated and unsaturated regions. Relevant examples arise in reaction-diffusion, elasto-plasticity, fluid dynamics, and so on.

Free boundaries are often related to discontinuities in constitutive relations; this raises a number of interesting analytical and numerical problems. Existence of the solution in function spaces, uniqueness, regularity properties, numerical approximation procedures, and other questions have extensively been investigated in the last decades. Several of these problems are of industrial interest, and offer opportunities of collaboration among mathematicians, physicists, engineers, material scientists, and other applicative scientists.

Models of hysteresis.

Hysteresis can be defined as a rate-independent memory. In physics we encounter it in plasticity, ferromagnetism, ferroelectricity, superconductivity, porous media filtration, and in many other phenomena. Several models of hysteresis phenomena have been proposed by physicists in the last two centuries; these include the theory known as micromagnetics, the Weiss theory, the Preisach model.

A systematic investigation of the mathematical properties of hysteresis from the viewpoint of functional analysis only began in the 1970s, when M.A. Krasnosel'skii introduced and systematically studied the concept of hysteresis operator acting between spaces of functions of time.

In the 1980s other mathematicians also began to study hysteresis phenomena, in particular in connection with PDEs and applicative problems. In the late 1990s a new viewpoint emerged, under the keywords of rate-independence and energetic formulation.

Two-Scale Models of Homogenization.

Continuum systems consisting of composite materials may be represented via P.D.E.s with rapidly-oscillating coefficients. Physicists, engineers and mathematicians have been dealing with these models via  asymptotic expansions for a long time. A new approach emerged in the 1970s with the seminal works of Babuska, De Giorgi and Spagnolo, J.L. Lions and others. This was then developed in a large literature under the keyword homogenization.

The point of view of multiscaling is of high relevance for applications, and was used for the homogenization of a number of models at the P.D.E.s issued from mathematical physics and engineering. A rigorous basis to two-scale models was provided by the notion of two-scale convergence, that was introduced by Nguetseng and then developed by Allaire about 20 years ago.

Control Problems.

Control problems are also studied. In particular, one of the research fields is the application of the dynamic programming technique and of the viscosity solutions theory of Hamilton-Jacobi equations, to optimal control problems for differential equations with hysteresis. These are interesting example of the so-called hybrid systems.

Staff