This group is mainly concerned with the study of nonlinear partial differential equations (PDEs) 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 indeterminacy.
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 by mathematical analysts in the last decades. Several of these problems are of applicative interest, and offer opportunities of collaboration among mathematicians, physicists, engineers, material scientists, and other scientists.
Models of hysteresis
Hysteresis can be defined as 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, and so on.
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 and his school 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. Discontinuous hysteresis was also addressed. 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 PDEs 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, Tartar and others. This was then developed in a large literature under the keyword homogenization.
The point of view of multi-scaling 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 more than 20 years ago.
Structural Stability of PDEs
This concerns the dependence of the solution of a PDE not only on the data, but also on the differential operator. A step in this direction has been done by the variational formulation of maximal monotone operators due to Fitzpatrick. This variational approach allowed for the application of De Giorgi's notion of Gamma-convergence to evolutionary PDEs, in particular quasilinear parabolic equations.
- Augusto Visintin - Professore senior