The research project focuses on PDE's arising, in particular, from problems in geometry or applied sciences. More precisely we will consider elliptic equations, both semilinear or fully nonlinear, and reaction-diffusion equations.
In the framework of elliptic equations arising in geometry 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 and other PDE's admitting a variational formulation and involving exponential nonlinearities;
-scale invariant problems and critical exponents for Pucci's operators.
Concerning PDE's relevant in population dynamics and other applied sciences we plan to study:
- reaction-diffusion equations;
- definition and properties of the principle eigenvalues and eigenfunctions and related issues for fully nonlinear elliptic equations, with singular or degenerate principal part, associated with mixed boundary conditions, for "truncated-laplacians" and for operators given by a fractional laplacian plus a transport term.
We believe that our research can lead to possible advances with respect to the current state of the art on the described themes. Moreover, as already pointed out and as we will further explain below, the topics under investigation have relevance in geometry, population dynamics and different other fields.
The problem of prescribing the Gaussian curvature on surfaces is a classical one for regular surfaces, but 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. We expect to show that in many cases the distribution of the singular points among the different connected components of the positive nodal regions 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, see [CLMP]. In this context uniqueness and multiplicity of solution depending on the potential is a widely investigated problem. In particular in some regimes [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.
The problem of prescribing the Q-curvature is relevant in geometrical analysis, but recently Q-curvature has also been studied in theoretical physics; in particular, to study anomalies in quantum field theory [N] and higher-derivative field theories [LO].
A part from its mathematical interest, overdetermined problems have great importance in applications, in particular in Shape Optimization Theory. Indeed they appear in the study of a viscous incompressible fluid moving in parallel streamlines through a straight pipe of given cross sectional form, in the linear theory of torsion of a solid straight bar of fixed cross section and also in the study of a liquid rising in a straight capillary tube of given cross section.
Also the study of solutions to curvature type equations (both local and nonlocal), besides its intrinsic relevance in geometry, is relevant in many applications. In particular the fully nonlinear operators considered appear in obstacle problems, deterministic and stochastic optimal control theory.
The implications of the study of reaction-diffusion equations in population dynamics, epidemiology, social sciences and other fields are self evident. It is worth to point out that the problems with a truncated Laplacian as diffusion operator would model diffusions that are only in the direction of some eigenvalues of the Hessian, i.e. the diffusion is in the directions where the growths is more decreasing or more increasing depending on the choice of the model. This type of degenerate problems has found many applications, in particular in image analysis.
Differential problems equipped with mixed boundary conditions play a significant role both from theoretical and applicative point of view. A very basic but enlightening example is that of a drum whose batter head may be attached to the hoop through "nails": the function that measures the oscillations of the batter head satisfies Dirichlet boundary conditions on the part of the boundary corresponding to the nails, and Neumann boundary conditions on the rest of the boundary. Moreover they are a prerequisite to answer the stability of the invasion if one wants to widen the scope of the theory of FKPP equations with Neumann boundary conditions to fully non-linear and also to non-uniformly elliptic operators.
At last operators given by the fractional laplacian plus a transport term naturally arise in some recent models in ecology and social sciences. In particular, the theoretical results could be applied to models on the impact of a climate change on biological populations.
[BJLY]Bartolucci-Jevnikar-Lee-Wang;JMPA2019
[CLMP]Caglioti-Liond-Marchioro-Pulvirenti;CommMathPhys1995
[LO]Levy-Oz;JHighEnergyPhys2018
[N]Nakayama;Phys.Rev.D2018