Research Interests
Nonlinear partial differential equations, numerical analysis, superconductivity, porous media, homogenization, calculus of variations, mathematical modeling
The Ginzburg-Landau theory of superconductivity describes the physical properties of interest for low temperature superconductors. Starting from the Ginzburg-Landau energy functional, the application of calculus of variations techniques gives rise to a system of semilinear elliptic partial differential equations (the Ginzburg-Landau equations). The study of generalized forms of the Ginzburg-Landau energy functional and of the systems which arise from them is an interesting open problem, both from a physical point of view, since there is an interest in extending such theory to the high temperature superconductors, and from a mathematical one for the difficulties in the analysis of the nonlinear equations derived
The presence in type II superconductors of a vortex structure (zeros of a complex valued functions, which represents the density of the superconducting electron pairs) is a source of engaging numerical problems, for the difficulties of construct efficient numerical algorithms and for the need of a deepened knowledge of high temperature superconductors, which are in particular of type II. From a computational point of view, an issue of interest is the numerical approximation of those simplified models which capture the essential characteristics of the vortex dynamics
Forchheimer law is a nonlinear equation which describes the flux in a porous medium under certain restrictions, in particular when the velocity of the fluid considered is above the range of validity of Darcy's law but turbulence is not yet appeared. Its generalization to the flux in a fractured porous material has practical interests in the field of oil extraction. Classically, homogenization is used in problems of this type, where the ultimate goal is to derive models in order to numerically approximate the phenomenon. Problems of convergence, existence and uniqueness of the systems so found require the use of sophisticated mathematical techniques