The group carries out teaching and research activities.
The teaching activity concerns courses in Mathematical Analysis and Mathematical Methods for degree courses at the various sites of the Faculty of Engineering.
The group’s research activity deals with various fields of Mathematical Analysis, in particular focusing on the study of the existence and multiplicity of solutions of ordinary and partial differential equations using variational and topological methods, degree theory and techniques for dynamical systems. Members of the group regularly organise conferences and seminars.
The group’s main lines of research are as follows:
Variational techniques are applied in the study of the problem of the existence and multiplicity of different types of integer solutions for elliptic semilinear equations or systems of equations that can be traced back to equations from Mathematical Physics such as Ginzburg-Landau, Allen-Cahn, Sine-Gordon and Schroedinger stationary.
The main objective is to study, both on bounded and unbounded intervals, the existence of solutions for boundary problems associated with differential equations. The equations studied may also be strongly non-linear, e.g. with Phi-Laplacian type terms, singular equations or functional equations. The boundary conditions may also be multi-point, integral or, more generally, functional.
The methods we usually use are of the topological type, i.e. fixed point theorems combined with above and below solution techniques.
Classically, both the topological degree theory of Brouwer and Leray-Schauder and the fixed point index theory find applications in various contexts. It is therefore of interest to extend these theories to more general classes of functions.
As an example, we have recently considered Fredholm maps of zero index between varieties in Banach spaces and their applications, for example in the field of differential equations on varieties or equations with delay. We also study various properties related to degree and index for maps in Banach spaces and the definition of eigenvalues for nonlinear maps. We have recently obtained an abstract theorem of existence of solutions in affine cones, similar to the Birkhoff-Kellogg theorem, and this result can be applied in different contexts such as equations with delay or with deviated argument.