Schedule for: 18w5209 - Physical, Geometrical and Analytical Aspects of Mean Field Systems of Liouville Type

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We consider traveling wave solutions to Gross-Pitaeskii equation with speed $c $ ($c \in (0, \sqrt{2})$. Using Adler-Moser polynomials and integrable system theory we construct high energy solutions when $c\to 0$. When $ c\to \sqrt{2}$ (the sound speed), the system is reduced to KP-I equation. Using Backlund transform at linearized level, we prove the nondegeneracy, Morse index and orbital stability of the KP-I lump solution. (Joint work with Yong Liu)

The problem of prescribing the Gaussian curvature on compact surfaces is a classic one, and dates back to the works of Berger, Moser, Kazdan & Warner, etc. Our aim is to consider surfaces with boundary where we also prescribe the geodesic curvature of it. This gives rise to a Liouville equation under nonlinear Neumann boundary conditions. In this talk we address the case of negative gaussian curvature. We study the geometric properties of the corresponding energy functional, and deduce the existence of minimum or mountain pass critical points. For that, a compactness result is in order. Here the cancellation between the area and length terms make it possible to have blowing-up solutions with infinite mass. This phenomenon seems to be entirely new in the related literature. This is joint work with Andrea Malchiodi (SNS Pisa) and Rafael López Soriano (U. Granada).

We show that the total area of two distinct Gaussian curvature 1 surfaces with the same conformal factor on the boundary, which are also conformal to the Euclidean unit disk, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total areas as the Sphere Covering Inequality. This inequality and it’s generalizations are applied to a number of open problems related to Moser-Trudinger type inequalities, mean field equations and Onsager vortices, etc, and yield optimal results. In particular we confirm the best constant of a Moser-Truidinger type inequality conjectured by A. Chang and P. Yang in 1987. This is a joint work Changfeng Gui.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Consider a Riemannian metric of constant curvature 1 with finitely many conic singularities on the sphere. Mondello and Panov recently described in the generic case what angles at the singularities are possible. We complete this description by finding the necessary and sufficient condition on the angles in the remaining case.

Adams’ inequality is the complete generalization of the Trudinger–Moser inequality tothe case of Sobolev spaces involving higher order derivatives. The failure of the original form of the sharp inequality when the problem is considered on the whole space R^n served as a motivation to investigate in the direction of a refined sharp inequality, the so-called Adams’ inequality with the exact growth condition. Due to the difficulties arising in the higher order case from the lack of direct symmetrization techniques, this refined result is known to hold on first- and second-order Sobolev spaces only. We extend the validity of Adams’ inequality with the exact growth to higher order Sobolev spaces. This is a joint work with Nader Masmoudi.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Liouville equations have interest in spectral theory, as they arise when extremizing so-called Functional Determinants. These are constructed out of spectra of conformally covariant operators, and are explicit in dimension two and four, due to formulas by Polyakov and Branson-Oersted. We discuss some existence, uniqueness, non-uniqueness results and some open problems.

Contrary to the Yamabe case, metric of prescribed Q-curvature in dimension 4 and higher, can blow-up both on isolated points and on higher dimensional submanifolds, as discovered by Adimurthi, F. Robert and M. Struwe. We will show, in a radially symmetric situation, that both kind of blow-up behaviour can happen at the same time. This is based on a joint work with A. Hyder.

This is joint work with D. Karmakar, C.-S. Lin and J. Wei. To each simple Lie algebra, there is associated a Toda system on the plane, with possible singularities at the origin. In this talk, we will discuss the classification of solutions to such Toda systems and the quantization result on their total masses. This generalizes various previous classification and quantization results for Lie algebras of type A, B and C. The total masses are related to the longest element in the Weyl group of the Lie algebra. Applications of these total masses to the local blowup masses of mean field equations will also be mentioned.

We are concerned with the global analysis of the unbounded solution branches of the Mean Field Equations (M.F.E.'s for short) which cross the critical value 8\pi. In particular, we will discuss some recent results where this long standing open problem problem is attacked by a careful analysis based on the interplay between M.F.E.'s and one of their underlying physical models, which is the Onsager statistical theory of 2d turbulence. It turns out that, to solve another open problem about the mean field entropy, one is bound to derive some new uniqueness and non degeneracy results of independent interest for blow up sequences of M.F.E.'s. This is a joint research project with A. Jevnikar, Y. Lee and W. Yang.

We consider the mean field equation \begin{equation*} \Delta_g u + \rho\left(\frac{h e^u}{\int_M h e^u}-\frac{1}{|M|}\right)=0 \end{equation*} where $h$ is a positive function, $(M,g)$ is a closed Riemann surface, $\Delta_g$ is the associated Laplace-Beltrami operator and $|M|$ is the total area of the surface. It is well known that nondegeneracy condition on a combination function $f_h$ involving $h$, the curvature and the Green function implies the existence of blow-up solutions as $\rho\rightarrow 8\pi m$ when $m$ is any positive integer. We construct blow-up solutions of the degenerate cases when the underlying manifold $M$ is a sphere or torus and $h\equiv 1$ by assuming some additional symmetry. This is a joint work with Ze Cheng (UTSA) and Changfeng Gui (UTSA).

We present results on the existence of (-1)-homogeneous axisymmetric solutions of incompressible stationary Navier-Stokes equations which are smooth on the unit sphere minus the north and south poles. In particular we classify such solutions with no swirl, and analyze their limiting behavior as viscosity tends to zero.

We deduce new uniqueness results for solutions to singular Liouville-type equations both on spheres and on bounded domains, as well as new self-contained proofs of previously known results. To this end, we derive a singular Sphere Covering Inequality based on the singular Alexandrov-Bol isoperimetric inequality and symmetric rearrangements. Work in collaboration with D. Bartolucci, C. Gui and A. Moradifam

On a convex bounded Euclidean domain, the ground state eigenfunction for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state eigenfunctions should have similar concavity properties. In this talk, I show that this is false, by analysing the perturbation problem from the Neumann case. In particular, we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground state eigenfunction is not log-concave (and indeed even has some super level sets which are non-convex) for small Robin parameter o polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance. The results presented in this talk are from my recent paper [arXiv:1711.02779] obtained in joint work with Ben Andrews (ANU in Canberra) and Julie Clutterbuck (Monash in Melbourne).

By a classical result of Kazdan-Warner, for any smooth sign-changing function f with negative mean on the torus there exists a conformal metric of Gauss curvature f, which can be obtained from a minimizer of Dirichlet's integral in a suitable class of functions. As shown by Galimberti, following our joint work with Borer on the higher-genus case, these minimizers exhibit ``bubbling'' in a certain limit regime. Here, we sharpen these results by showing that all ``bubbles'' must be spherical. Moreover, we revisit Galimberti's result and prove that analogous ``bubbling'' occurs in the prescribed curvature flow.

Motivated by the study of non-abelian Chern Simons vortices of non-topological type in Gauge Field Theory we analyse the solvability of some Liouville-type system in presence of singular sources. We identify necessary and sufficient conditions which ensure the radial solvability of this system.

We consider the classical Lane Emden equation in bounded domains of the plane with Dirichlet boundary conditions and we present some results concerning the Morse index of solutions to this problem, when the exponent of the nonlinearity is large. Via these Morse index computations and a precise asymptotic analysis we can deduce a uniqueness result for positive solutions in convex domains and also some existence results of non-radial sign-changing solutions in the ball. Based on joint papers with M. Grossi, I. Ianni and F. Pacella.

We discuss existence issues for a Dirichlet problem in bounded domains with polynomial nonlinearities of critical growth in presence of an Hardy potential. Linear perturbations can produce positive solutions and we aim to construct sign-changing solutions, shaped as a tower of "bubbles" centered at 0 with alternating signs. The construction is optimal in the radial case as a very fine asymptotic analysis shows for radial towers of "bubbles", recovering and improving what is known in the non-singular case. Joint work with N. Ghoussoub, A. Pistoia and G. Vaira.

This talk is concerned with the singular mean field problem of Liouville type on compact surfaces. The study of this equation is motivated by some problems which arise in Differential Geometry and Physics. We will focus on the existence and compactness of solutions, which have been extensively studied for positive potentials. However, the case of sign changing potentials has not been much considered in the literature. For the latter case, we present results on the solvability using variational techniques. Concerning the compactness, we deal with the possible blow--up phenomena in order to establish a general criterion. Joint works with Francesca De Marchis and David Ruiz.

The \emph{carr\'e du champ} method and its nonlinear counterpart is a powerful technique to prove uniqueness in some nonlinear elliptic PDEs and establish optimal constants in interpolation inequalities or optimal rates of decay in related evolution problems. It is related with entropy methods in PDEs inspired by generalizations of Boltzmann's entropy. Optimality cases can be identified by considering asymptotic regimes and appropriate linearizations. The method applies to symmetry results in presence of weights. It does not rely on symmetrization but raises various issues of regularity and rely on integrations by parts which are not always straightforward to justify. This lecture will be devoted to an overview of the main results.

I will consider a system of two coupled Liouville equations on the plane. The system admits so-called scalar solutions, namely such that the two components coincide. These solutions actually solve a scalar Liouville equation on the plane, hence they are very well known and they have been completely classified. On the other hand, much less is known about non-scalar solutions. Using bifurcation theory, I will show the existence of some branches of (non-scalar) solutions bifurcating from a scalar solution.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.