Many physical principles are an expression of an underlying variational principle; Euler made the expansive statement that
``...nothing at all takes place in the universe in which some rule of the maximum or minimum does not appear.''
When the maximum or minimum is attained -- for example, when potential energy is minimized -- the physical law is also described by a partial differential equation, the Euler-Lagrange equation.
At the same time, physical laws often have a purely geometric description. Einstein's field equations of General Relativity prescribe the Ricci curvature tensor of space-time. Soap bubbles minimize area and therefore, in physical terms, are the least energy configuration.
This property is geometrically expressed by the fact that soap bubbles have constant mean curvature. However, the roundness of soap bubbles was proved by Alexandrov
using PDE techniques, i.e., by analyzing the Euler-Lagrange equation. Therefore, one is often compelled to view physical phenomena while wearing three hats: that of the physicist, the geometer, and the analyst.
The examples of the Einstein field equations and minimal surfaces are particularly instructive, because their study drove many of the developments in the existence
and regularity theory for nonlinear PDEs in the last century. The goal of our proposal will be to bring together researchers who are working PDEs with connections to differential geometry and mathematical physics.
We again discuss separately the three cases described in the previous sections
(P1) Concerning the classification of the solutions to the system (T), the proof in [JW] strongly relies on the algebraic invariance of the Cartan matrix. For other type of matrixes these invariants are not available and it is an interesting problem to characterize the matrices for which a solution does exist.
Since the problem is not variational, the bifurcation theory seems to be a powerful and promising tool.
Concerning the blow-up phenomena for a more general system (T), it would be interesting to prove that all the blow-up values found in [BP] are the only possible values for the local masses.
It would be also interesting to explore the existence of solutions
whose blow-up behavior includes also a certain global mass (see e.g. [DPR]).
[BP] L. Battaglia, A. Pistoia A unified approach... arXiv:1607.00427
[DPR] T. D'Aprile, A.Pistoia, D.Ruiz, Asymmetric ... JFA 2016
[JW] J. Jost, G. Wang, Classification...IMRN 2002
(P2) We will consider system (S) with critical growth in a fully inhomogeneous medium that we model by an arbitrary compact Riemannian manifold, thus breaking the various symmetries that these systems enjoy in the Euclidean setting. We want to study the critical system in the spirit of [DH,DHV,EPV]. In particular, we are interesting in building blowing-up solutions: the choice of the building block is the most delicate part (see e.g. [EPV]).
The phenomenon of spatial segregation in dynamic of populations will also be considered. As a prototype for the study of this phenomenon, a competition-diffusion system of k differential equations was considered in [CTV1]. The case of two or three populations was studied in details in [CTV2,CTV3].
We intend to study the problem of the uniqueness of the limit configuration and its qualitative properties in the case of four species.
[DH] O.Druet, E.Hebey, Stability for strongly ... Anal. PDE 2009
[DH] O.Druet, E.Hebey, J.Vétois, Bounded stability ... JFA 2010
[CTV1] M Conti, S. Terracini, G. Verzini, A Variational Problem ...IUMJ 2005
[CTV2] M Conti, S. Terracini, G. Verzini, Asymptotic estimates ... Adv. Math. 2005
[CTV3] M Conti, S. Terracini, G. Verzini, Uniqueness and least ... Interfaces and free boundaries 2006
[EPV] P.Esposito, A. Pistoia, J. Vétois, The effect of linear ... Math. Ann. 2014
(P3) We plan to give upper and lower bounds of the magnetic Laplacian in terms of the geometry of the manifold and the fluxes of the potential around the integral homology classes. This operator combines the geometry and the combinatorics of the set of fluxes of the potential. The novelty is in an explicit form of the lower bound, which apparently does not exist in the literature. First results (in incomplete form) have been obtained in [CS].
We also plan to study the spectrum of the Dirichlet-to-Neumann operator, with special attention to the asymptotics of its counting function (which, at the positive number x, measures the number of eigenvalues which are less than x). Particularly intriguing, and new, would be its relation with the Laplacian with Robin boundary conditions. This problem is a natural generalization of the classical Dirichlet-to-Neumann (or Steklov) problem on functions (see [RS]).
[CS] B. Colbois, A. Savo Eigenvalue bounds¿ arXiv:1611.01930
[RS] S. Raulot, A. Savo, On the first eigenvalue... JFA 2012