The present research project focuses on some specific topics arising in the theory of nonlinear elliptic partial differential equations, ranging from the uniformly elliptic regime to some strongly degenerate cases.
More precisely, we plan to study the following problems:
1) Uniformly Elliptic Equations, with special attention to
-Radial critical exponents for Pucci's operators;
-Morse index for radial sign-changing solutions of classical Lane-
Emden and Hénon equations;
-Overdetermined elliptic problems;
- Critical points of Moser-Trudinger functional on surfaces.
2) Systems of elliptic equations, in particular including
- Fully nonlinear systems with Pucci's operators;
- Reaction-Diffusion Systems in population dynamics.
3) Nonlocal and degenerate elliptic equations, focusing on
- Fractional truncated Laplacians;
- Mixed boundary problems for degenerate/singular equations;
- Inequalities for pseudo-differential operators.
The methodology we intend to implement involves different techniques, from a dynamical systems approach for radial uniformly elliptic equations to comparison principles and ad-hoc constructed barrier functions for the case of strongly degenerate operators, passing through bifurcation methods for the variational cases.
The proposed research can lead to consistent advances with respect to the current state of the art in the theory of nonlinear elliptic equations. Moreover, the topics under investigation have relevance in applied models and/or play a role in neighboring fields. The proposed research will be carried on by means of already well established collaborations of the participants with international young researchers and leading experts in the field.
We believe that our research can lead to possible advances with respect to the current state of the art on the described themes. Moreover, the topics under investigation have relevance in applied models and/or play a role in neighboring fields.
In particular, in the theory of fully nonlinear elliptic equations, the knowledge of the critical exponents for Pucci's operators, though limited to the radial case, can bring to sharp existence results in general bounded domains. The radial analysis at the critical level, carried on despite the lack of linear and variational structure, reflects intrinsic properties of the operators, and a kind of a related energy invariance, which could be applied beyond the radial setting.
The analogous results for systems of fully nonlinear equations would bring to a consistent progress of the theory as well, since no exact threshold for existence of solutions is known up to now.
Likewise, new results for reaction-diffusion systems in population dynamics with a large number of competing species would be far-reaching for applications.
As for the analysis of the Moser-Trudinger functional is concerned, we recall that the study of the problem on planar domains dates back to [CC] and [F], where the existence of a global maximizer was proved for a threshold value of the Sobolev norm, while in the subcritical case the existence of a global maximizer is trivial. Later, in some supercritical ranges, critical points of saddle type were detected e.g. in [LRS] and [DMR] and more recently a degree formula has been announced for the functional on planar domains and generic values of M. Even if the same approach could work in the surfaces' setting, it would not imply any existence of critical points on the sphere.
The proposed research for nonlocal strongly degenerate elliptic equations is completely new. We recall that the fractional laplacian is used to model many problems arising in different frameworks, such as optimization [DL], probability [Be], finance [CT], phase transitions [SV], stratified materials [SaVa], anomalous diffusion [M], crystal dislocation [N], the ultrarelativistic limit of quantum mechanics [FDL], quasigeostrophic flows [C]. The fractional truncated laplacian could be used to analyze refined models for more complex systems with degeneracy.
REFERENCES:
[Be] Bertoin, Cambridge Tracts in Mathematics(1996)
[CC] Carleson, Chang, Bull. Sci. Mat. (1986)
[CT] Cont, Tankov, Chapman & Hall/CRC, Boca Raton, FL(2004)
[C] Cordoba, Ann. of Math. (1998)
[DMR] del Pino, Musso, Ruf, Calc. Var PDE (2012)
[DL] Duvaut,Lions , Springer-Verlag, Berlin (1976)
[FDL] Fefferman, De La Llave, Rev.Mat. Iber.(1986)
[F] Flucher, Comment. Math. Helv. (1992)
[LRS] Lamm, Robert, Struwe, J. Funct. Analysis (2009)
[M] Metzler, Klafter, Phys. Rep. (2000)
[N] Nabarro, Adv. in Physics (1952)
[SaVa] Savin, Valdinoci, J. Geom. Anal.(2009)
[SV] Sire, Valdinoci, J. Funct. Anal. (2009)