The research project focuses on some nonlinear PDE¿s arising, in particular, from problems in Geometry or Applied Sciences. More precisely we will consider elliptic equations, both semilinear or fully nonlinear, Hamilton-Jacobi equations and nonlinear conservation laws.
In the framework of elliptic equations we will study :
- overdetermined problems to the aim of characterizing the domains for which a solution exists.
- the question of prescribing Gaussian curvature under conformal changes of metrics admitting conical singularities
- fully nonlinear elliptic equations which involve degenerate, singular or uniformly elliptic operators.
Concerning Hamilton-Jacobi equations we plan to study :
- ergodic problems in noncompact settings
- discontinuous viscosity solutions on graphs and networks
- Hamilton Jacobi equations with discountinuous initial data
As far as conservation laws are concerned, we will study :
- measure valued solutions for scalar conservation laws
- propagating fronts for hyperbolic reaction-diffusion equations
We believe that our research project can contribute to interesting advances in each field of investigation, sometimes by innotive mehods.
ELLIPTIC EQUATIONS
The study of overdetermined problems in domains with a free boundary inside another set is a quite new area of research. In [PT] the case of domains inside a cone is studied, but other geometries should be considered to detect the right boundary conditions which determine the shape of a domain. Analogously the study of constant mean curvature surfaces seen as free boundary of domains has been very little investigated. Finally the connections with isoperimetric inequalities are only partially understood, therefore the construction of adhoc examples could give new intuitions about the shape of sets minimizing the relative perimeter.
The problem of prescribing the Gaussian curvature on singular surfaces is considerably more difficult than the regular one and, in particular, the question of analyzing sign-changing curvatures and singularities with orders of mixed sign is completely open. We expect to show that 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.
Concerning fully nonlinear equations we believe that the critical radial exponents for Pucci¿s operators reflect intrinsic properties of the operators, not limited to radial solutions. We wish to make a precise analysis of the radial case, through integral characterizations of the critical exponents and explicit estimates of the solutions. This could lead to the construction of solutions in arbitrary domains by perturbative arguments, as for domains close to balls [EFQ].
Concerning the ergodic problem, its analysis leads, in particular, to the construction of solutions blowing up on the boundary. The asymptotic expansions of these solutions involve the operator evaluated on the second fundamental form of the boundary, which shows the interplay between the operator and the geometry of the domain for the existence of solutions. Moreover, the analysis of the ergodic problem requires a priori Lipschitz estimates and comparison principles for degenerate elliptic equations which are results of independent interest.
Finally we believe that the study of the truncated Laplacian is very interesting because its strong degeneracy reflects in the lack of validity of Harnack and ABP inequalities, maximum principles, Green representation formulas, etc. This yields the occurrence of new phenomena with respect to the Laplacian. The new arguments introduced to overcome this pathology could be used to study the existence and symmetry of eigenfunctions in unknown cases, the simplicity of first eigenvalue, as well as to analyze the "singular" counterpart of the works [BGL], [G], i.e. equations with zero order terms with negative powers.
HAMILTON-JACOBI EQUATIONS
Concerning ergodic problems our investigation can be framed in the study of possible generalizations of Weak KAM and Aubry Mather theory beyond the compact case. It is a widely open field, see [C01], where a compactification of the Aubry set at infinity is proposed, and [DS1, DS2, DS3, DS4], [II]. In particular the generalization to second order equations seems connected to the determination of representation formulae easy to handle.
Morever discontinuous viscosity solutions for Hamilton-Jacobi equations appear in mean fields theory of current interest. Their study in discrete settings, as networks and graphs is however at an embryonal stage, see [CM].
CONSERVATION LAWS
For measure-valued solutions, the innovative part consists in determining a formulation in a context with a high degree of freedom in the choice of the initial measure and of the flux. The case of measures with variable sign has been addressed only in the case of fluxes where the interaction between measures of different sign does not take place. Furthermore qualitative aspects of solutions with bounded fluxes need to be studied.
Concerning propagating fronts, the available results are limited to a reactive variation of the two-speeds Goldstein-Kac model and almost no result is available when the speeds are more than two. New outcomes concern the existence of a domain of attraction of propagating fronts for hyperbolic reaction-diffusion equations for a new class of weakly coupled systems with discrete velocities. In addition, an explicit formula for the critical speed is expected to be rigorously proved in the logistic case.
REFERENCES
[BGL] Birindelli, Galise, Leoni, Nonlin. Anal. 2017
[CM] Camilli, Marchi; SICON 2016
[C01] Contreras; CVPDE 2001
[DS1] Davini, Siconolfi; Math. Ann. 2009
[DS2] Davini, Siconolfi; CVPDE 2012
[DS3] Davini, Siconolfi; CVPDE 2011
[DS4] Davini, Siconolfi; Ann. IHP 2016
[EFQ] Esteban, Felmer, Quaas, CPDE 2007
[G] Galise, JDE 2019
[II] Ichihara, Ishii, CPDE 2008
[PT] Pacella, Tralli, Rev. Mat. Iberoamer (to appear)