The project focuses on the study of some classes of nonlinear elliptic equation which arise in Geometry and Applied Sciences. The aim is to analyze the following questions :
1) Surfaces with constant mean curvature and related isoperimetric inequalities.
2) The problem of prescribing the Gaussian curvature on surfaces with conical singularities.
3) Qualitative properties of positive solutions of Lane-Emden problems in planar convex domains.
4) Asymptotic behavior and concentration phenomena of solutions of fully nonlinear elliptic equations involving the Pucci operators in a ball.
5) Regularity results for solutions of a class of degenerate fully nonlinear elliptic equations.
6) Elliptic and parabolic problems with a Hardy potential.
The project focuses on various open 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.
SEMILINEAR ELLIPTIC EQUATIONS
As stated in the project we plan to prove uniqueness of the positive solution of Lane Emden problems in planar convex domains in the case the nonlinear term has a large exponent. This would be a significant step in the direction of proving the complete uniqueness result. Indeed, since when the exponent is small the uniqueness result easily follows, it would remain to analyze only the case when the exponent lays in a bounded interval and , probably , a fine analysis of the solutions branches could lead to the full uniqueness result.
The problem of prescribing the Gaussian curvature on surfaces is a classical one, solved by Kazdan-Warner in [Ann. Math 1974 ] for regular surfaces. The case of surfaces with conical singularities is considerably more difficult and in particular the question of analyzing conical singularities of negative orders or of orders of mixed sign is completely open.
FULLY NONLINEAR ELLIPTIC EQUATIONS EQUATIONS.
We plan to show that a class of fully nonlinear equations involving Pucci's extremal operators exhibits a concentration phenomenum when the exponent of the nonlinearity approaches a critical value.
This result would be the first one of this type for the equations under consideration. Its interest also lies in the fact that the problems we consider are not variational and so far concentration results have only obtained for solutions of equations having a variational structure.
Concerning the second topic let us point out that the kind of equations we consider involve higly degenerate fully nonlinear operators, namely the "truncated Laplacians" which have not been much studied in the literature, though they play an important role in Geometry and also appear in the theory of stochastic control. To prove the Lipschitz regularity of the solutions for all values of k (see the research proposal) will represent a good progress in the theory of the regularity of viscosity solutions of fully nonlinear equations.
CONSTANT MEAN CURVATURE SURFACES
The classical result of Alexandrov for constant mean curvature surfaces applies to compact surfaces without boundary. In the case of surfaces with boundary several results have been obtained in cases when the moving plane method can be applied giving as result symmetric surfaces.
We plan to consider constant mean curvature surfaces with boundary which do not have any symmetry but lie in cones, the result we expect should be that they are portion of spheres but with no symmetry, unless the cone itself has some symmetry. Cases of this type are not present in the literature. Moreover the result we want to get would be related to some relative isoperimetric inequalities obtained by P.L. Lions and F.Pacella only in the case of convex cones. We believe that the result on constant mean curvature surfaces that we want to get should allow to extend the relative isoperimetric inequality to other kind of cones for sets whose boundary is a graph.