# Asymptotic Patterns in Variational Problems: PDE and Geometric Aspects (16w5065)

Arriving in Oaxaca, Mexico Sunday, September 25 and departing Friday September 30, 2016

## Organizers

(Universität Giessen)

(Universitá di Torino)

(University of Sydney)

(Sapienza Università di Roma)

## Objectives

In many models of applied science and problems from other areas of mathematics, in particular differential geometry, one observes the formation of singularities or patterns as a parameter approaches a singular value. Typical problems of this kind are systems of PDEs with competitive interaction when the interaction parameter becomes large, nonlinear Schr\"odinger equations in the semi-classical limit, Bose-Einstein condensation in different spin states, but also problems from conformal geometry where the parameters are hidden. In order to understand these phenomena a key issue is to identify an associated singular limit problem and to investigate its solution structure. This can be an optimal partition problem, an autonomous PDE or a system in entire space, or an ODE.

A special and fascinating feature of these problems is the interplay between analysis and geometry, between solutions of PDEs and trajectories of ODEs. It is the goal of this workshop to bring together specialists from various sides of this topic for an exchange of ideas about the PDEs, the limiting problems, and the passage between them.

We now describe the main topics in detail. We would like to emphasize the remarkable unity in methodology running across the different topics of the proposal. Indeed, they already share many common techniques and, in our intentions, will further benefit from the exchange of attack strategies resulting from this workshop.\\

1. Interacting systems and optimal partition problems\\

Reaction diffusion systems with strong competitive interactions appear in several natural and physical phenomena and can be described by a certain number of densities distributed in a domain, subject to diffusion and competition interations. Whenever the competitive interaction is the prevailing phenomenon, the densities cannot coexist and tend to segregate, hence determining a partition of the domain. In this case, the partition becomes a major object of investigation, both from the analytical point of view and from the geometric one, with emphasis on the points of multiple intersection. When the system possesses a variational structure, one can associate an optimal partition problem with the ground states. Conversely one can regard optimal partitions related to linear or nonlinear eigenvalues as limits of competing systems problems as the competition parameter diverges.

A notable example of this kind of problem is the Gross-Pitaevskii system, which describes the solitary waves of a system of nonlinear Schr\"odinger equations appearing in condensation and superfluidity phenomena with different spin states. The effects of interparticle interactions are of fundamental importance in quantum mechanics. Other problems of this type come from population models for two or more competing species.

To start with, we consider the simplest, yet highly nontrivial case of stationary elliptic systems with many components; we deal with solutions to the system: $\begin{cases} \Delta u_i= u_i\sum_{j\neq i} u_i^2\\ u_i> 0 \quad \mbox{in} \quad \mathbb{R}^N\,. \end{cases}$ A main issue is to classify all possible entire solutions, depending on their algebraic growth rate. At this moment only partial results are available. More generally, we focus on a systematic classification of the entire positive solutions to the cubic Schr\"odinger System $\Delta u_i=u_i\sum \beta_{ij}u_j^2\;,$ relative to the properties of the matrix interspecific scattering lengths $(\beta_{ij})$. Solutions to the evolutionary parabolic problem connecting two entire solutions are of interest as well, together with the evolution of the full dime dependent Schr\"odinger system. and the stability of standing waves. The same problems are extremely interesting also in the case of nonstandard diffusions and non local interactions. These problems are technically very difficult and require a joint use of methods from geometrical PDE's and the theories of phase transitions and free boundaries.\\

We intend to bring together people working on Schr\"odinger equations and systems (stationary and time depending) and people working on optimal partition problems.\\

2. Entire solutions of semilinear elliptic equations and systems\\

Entire solutions of semilinear elliptic equations where no explicit dependence on space variables is present arise in many applications, a prototype being the Semilinear Poisson equation $$\Delta u + f(u)=0\quad\hbox{in }\mathbb{R}^N. \label{pro}$$ In many studies, problems like (1) are considered involving explicit dependence on the space variable, or on a manifold or in a domain in $\mathbb{R}^N$ under boundary conditions. Topological and geometric features of the domain are often characteristic that trigger the presence of interesting solutions, whose precise features can be analyzed when some singular perturbation parameter is involved. In the absence of space inhomogeneity or geometry of the ambient space, as in the clean'' equation (1), it is less clear which internal mechanisms of the equation are behind complex patterns in the solution set, whose richness may be nearly impossible to fully grasp.

Two natural problems, classical in the PDE literature for entire solutions are, on the one hand, that of classifying solutions of (1) when the solutions are assumed to satisfy additional properties: This is the context of various classical results in the literature like the Gidas-Ni-Nirenberg theorems on radial symmetry, Serrin's overdetermined problems, Liouville type theorems under assumptions such as stability or finite Morse index, or the achievements around De Giorgi's conjecture. In those results, the geometry of level sets of the solutions turns out to be a posteriori very simple (planes or spheres). On the other hand, problems of the form (1) for nonlinearities recurrent in the literature, do have solutions with interesting shapes. Constructing solutions, typically depending on a parameter, exhibiting meaningful asymptotic patterns is a challenging problem for which important results have appeared in recent years. This has been the case for instance in the Allen Cahn equation in phase transitions, the standing wave problem for nonlinear Schr\"odinger equations, Yamabe equations and Liouville type equations. Connected with this is the issue of overdetermined problems like Serrin's problem in unbounded domains, and the analysis of extremal or stable solutions of Gelfand type problems in bounded domains.

In recent years the above issues have also been considered for quasilinear or fully nonlinear elliptic problems. A very interesting direction has been the analysis of nonlocal operators, most notably, semilinear equations involving the fractional Laplacian. In particular the fractional Schr\"odinger and Allen Cahn equations, and the associated version of De Giorgi's conjecture. Deep results such as uniqueness of fractional ground states have been recently discovered. Connected with this is the new geometric subject of fractional minimal surfaces. Other direction in which important progress have been achieved are the classification and construction results involving polyharmonic operators, and systems modeling multiple phase separation, Toda type systems in 2d. In all those problems many important new technical challenges and new phenomena arise.

An important point to be explored is the connection between the above purely PDE issues and similar phenomena present in the theories of minimal and constant mean curvature surfaces and time-depending geometric flows. Topics such as singularity formation, infinite or finite time blow-up are also of interest. The dialog among people working on different sides of these matters, both the geometric and purely PDE matters is crucial for further advance, and in fact a major goal of the proposed workshop.\\

3. Concentration phenomena\\

Concentration phenomena have been the subject of many studies in the last two decades and the subject has developed enormously. Notable examples broadly treated in the literature are:
• nonlinear Schr\"odinger equations in the semi-classical limit or systems of such equations,
• systems modelling biological pattern formation based on Turing's morphogenesis theory such as the Gierer-Meinhardt model,
• the Keller-Segel model of chemotaxis,
• the gradient theory of phase transitions by Allen-Cahn and Cahn-Hilliard, and
• various related variational models in material science,
• the Ginzburg-Landau model for superconductivity,
• Gelfand's problem in combustion theory.
Often in the above settings one deals with a semilinear elliptic equation depending on a small parameter, defined in a region of Euclidean space or a finite dimensional Riemannian manifold, subject to an appropriate set of boundary conditions. Concentration phenomena have also been central in conformal geometry, where singular behavior arises in the form of bubbling triggered by the presence of critical behavior in the nonlinearities. Here the parameters come from scaling invariance and are somewhat hidden''. Of particular interest will be systems with a large parameter in all equations where the problem cannot be reduced to a system with only one equation singularly perturbed. These systems are not "almost scalar" and tend to be more difficult to study.

When the parameter tends to the singular value, solutions concentrate on subsets of the domain, and it is of fundamental importance to localize these subsets. Often the subsets can be described as critical sets of a limit function, defined on a finite-dimensional space and involving the Green's function of the domain. As an example, semilinear equations on two-dimensional domains with exponential type nonlinearity, e.g. the sinh-Poisson or the Liouville equation, lead to the $N$-vortex Hamiltonian from fluid dynamics as limit function. This is defined on the configuration space of $N$ points in the domain. Its critical points are stationary point vortex distributions, and solutions of the original PDE yield regularized vorticity fields for the Euler equation.

This workshop is partially sponsored by ERC Advanced Grant 2013 Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT(travel-related expenses)