Talk 1 (3pm):
Jose Antonio Carrillo de la Plata(Imperial College): Nonlinear Aggregation-Diffusion Equations in the Diffusion-dominated and Fair competition regimes
Abstract:
We analyse under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction, occurs. I will discuss several regimes that appear in aggregation diffusion problemswith homogeneous kernels. I will first concentrate in the fair competition casedistinguishing among porous medium like cases and fast diffusion like ones. I will discuss the main qualitative properties in terms of stationary states and minimizers of the free energies. In particular, all the porous medium cases are critical while the fast diffusion are not. In the second part, I will discuss the diffusion dominated case in which this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as time tends to infinity. This talk is based on works in collaboration with S. Hittmeir, B. Volzone and Y. Yao and with V. Calvez and F. Hoffmann.
Talk 2 (4.30pm):
Jean Lagace (UCL), On recent approaches towards Pólya’s conjecture
Abstract :
Eigenvalues of the Laplacian on a planar domain of unit area with either Dirichlet or Neumann boundary conditions form a sequence accumulating only at infinity. Pólya conjectured in 1962 that the k^th Dirichlet eigenvalue is bounded below by 4πk, while the k^th Neumann eigenvalue is bounded above by the same constant. The conjecture is known to be true for domains that tile the plane, and is known to hold for the first few eigenvalues, or for k large enough, but in no other situations, even the disk. The conjecture is notoriously hard to study in the intermediate regime, notably because we don’t know whether or not optimising domains for the k^th eigenvalue exist amongst planar domains, so we cannot use properties of an extremiser. I will present two new approaches : the first one restrains the problem to a class of domains amongst which extremisers are known to exist. The conjecture then becomes equivalent to some specific properties of these extremmisers holding. The second approach relates the Neumann problem to the Steklov problem, and obtains universal upper bounds for the former in terms of the latter. Based on joint work with Pedro Freitas (Lisbon) and Jordan Payette (Tel-Aviv), and with Alexandre Girouard (Laval) and Antoine Henrot (Nancy)