Ricerc@Sapienza

The project focuses on various questions whose (even partial) answers would represent major improvements of already available results. We support this claim by listing some of the main features of the project for each field of investigation.

LIOUVILLE TYPE PROBLEMS

The problem of prescribing the Gaussian curvature on surfaces is a classical one, solved by Kazdan-Warner [KW] for regular surfaces. The case of surfaces with conical singularities is considerably more difficult and in particular the question of analyzing sign-changing curvature and singularities with orders of mixed sign is completely open. Furthermore we expect to show that in many cases the distribution of the singular points among the different connected components of the positive nodal region of the curvature influences the solvability and this would be a completely new phenomenon.
As already mentioned the Liouville type equation appearing in this problem is also relevant in statistical mechanics, being a mean field equation for the Euler flow. In this context uniqueness and multiplicity of solution depending on the potential is a widely investigated problem. In particular in some regimes a recent result [BJLY] provides sufficient conditions which guarantee the uniqueness of the solution. By obtaining some multiplicity result if we assume some degeneracy of the potential, we could show that the assumptions in [BJLY] are "almost" necessary.
Finallly, concerning the problem of prescribing the Q-curvature, to remove the assumption of the nontriviality of the Kernel of the Paneitz operator will represent a significant progress.

CMC SURFACES

The study of CMC surfaces is one of the main topics in classical differential geometry [L]. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillarity phenomena. There are very few results for compact surfaces with boundary without prescribing the boundary itself, therefore the questions analyzed in the project should be of great interest. Moreover there is a strong connection between this question and the relative isoperimetric problem studied in [LP] which is also important in the capillarity theory. The corresponding partial overdetermined problem has been first addressed in [PT] and it has some significant applications in potential theory as for the classical Serrin's problem.

LANE-EMDEN PROBLEMS

As stated in the project we plan to study multipeak positive solutions, computing their degree and showing possibly their uniqueness for a given number of peaks when the exponent p is sufficiently large. This would be a significant step in the direction of drawing a bifurcation diagram for finite energy solutions.
At last a deep understanding of the scalar case could help in the study of the asymptotic behavior of solutions of systems made by (LE) equations; in particular it would be intersting to see if singular systems could be identified as limit problems.

COMPETITION-DIFFUSION SYSTEMS

One question we intend to analyze is the geometry description of the limit configuration in the case of four or more species. Another question concerns the study of a competition-diffusion system with the introduction of drift terms. Together with the main question of existence and uniqueness of the solution, our goal is to investigate the regularity properties of the limit configuration.

[BJLY]Bartolucci,Jevnikar,Lee,Wang;J.Math.PuresAppl.,to appear
[KW]Kazdan,Warner;AnnMath1974
[L]Lopez;book2013
[LP]LionsPacella;ProcAMS1990