Anagrafe della ricerca

The aim of this project is to develop the theoretical analysis of PDEs and related optimization issues recently emerging from several new directions. Our ultimate goal is to bring together highly motivated researchers willing to employ their competences to tackle theoretical, and application-oriented challenging problems. The project will be carried out according to the following lines of research: 1. GLOBAL ANALYSIS OF PDEs: nonlinear elliptic or parabolic, nonlocal, gradient flows, in non-euclidean frameworks (manifolds, graphs, metric spaces). 2. QUALITATIVE PROPERTIES OF SOLUTIONS: geometry, shape, regularity, asymptotics, for elliptic and evolution equations (e.g., aggregation-diffusion, anisotropic parabolic, reaction-convection-diffusion equations). 3. QUANTITATIVE FUNCTIONAL INEQUALITIES: Sobolev, Poincaré, Hardy-type, Brunn-Minkowski, Borell-Brascamp-Lieb inequalities; in euclidean and non-euclidean frameworks (manifolds, graphs, Finsler metrics). 4. OPTIMIZATION ISSUES AND APPLICATIONS: spectral functionals optimization (with respect to shape or weights) for second and fourth-order elliptic operators; first and second-order free-discontinuity problems; related applications, e.g., to mathematical models for bridges. The above topics have been thought to be closely interconnected in such a way that approaching them with synergy will bring several mutual benefits. As a matter of example, the planned global and qualitative analysis of diffusion problems (items 1-2) will benefit from the study of related elliptic problems, e.g., existence results for semilinear PDEs will have significant consequences for the well-posedness of degenerate diffusions. On the other hand, the development of suitable functional inequalities (item 3) will be fundamental for some of the proposed researches (items 1-2), e.g., in the study of the geometry of solutions or to appropriately set up anisotropic problems; vice versa, suitable PDE tools will be employed to prove sharp functional inequalities. Similarly, the optimization issues of item 4, despite being mostly motivated by applications, will bring challenging theoretical problems, e.g., for higher order operators or functionals; on the other hand, the analytic and geometric arguments developed in items 1-3, may be exploited in a number of applied models, see the detailed description below. To conclude, we stress that a common thread of the project will also be the analysis of PDEs and inequalities in non-euclidean frameworks, such as manifolds or graphs. Clearly, such analysis will benefit from the tools developed in the euclidean setting and the expertise gained by the components of the project in more classical contexts. Furthermore, working in nonstandard frameworks will lead to further insights, even in the euclidean case, by pointing out hidden connections and deeper dynamics of the equations, relating them to intrinsic properties of the ambient space.