Recent webinars

Europe/Lisbon
Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa — Online

Carlos N. Rautenberg

Carlos N. Rautenberg, George Mason University
Variational Problems on Measure Spaces with Constraints

We consider a class of variational problems involving partial differential operators on non-standard vector-valued measure spaces with divergences (represented by measures or by functions). This class of problems arise from the study of the growth of heterogeneous sand piles and via (Fenchel) duality of the aforementioned models. Further, mixed boundary conditions are in place and these are established by means of a normal trace characterization of the vector measure. We determine existence of solutions, stability and strong duality results. We finalize the talk with numerical tests.

Europe/Lisbon
Room P3.10, Mathematics Building — Online

Matthias Hofmann, Texas A&M
Computing eigenvalues of the discrete $p$-Laplacian via graph surgery

We discuss the dependence of the eigenvalues and eigenfunctions for the discrete signed $p$-Laplacian under perturbation by a cut parameter. In particular, we prove a formula for the derivative of the eigenvalues and show that the eigenvalues of the discrete signed $p$-Laplacian on the original graph can be characterized via extremal points of the perturbed system. In this context, we elaborate on how graph surgery can be used in order to compute eigenvalues of the discrete (signed) $p$-Laplacian by looking at some examples. The derivation formula is reminiscent of the formula for linear eigenvalue problems given by the Hellmann-Feynman theorem and our results extend previous results for the linear case $p=2$ attained by [Berkolaiko, Anal. PDE 6 (2013), no. 5, 1213-€“1233].

Europe/Lisbon
Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa — Online

Damião Araújo

Damião Araújo, Universidade Federal da Paraíba
Regularity in diffusion models with gradient activation

In this talk, we discuss sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process drives the system only in the region where the gradient surpasses a given threshold. This is joint work with Aelson Sobral, Universidade Federal da Paraíba - Brazil, and Eduardo Teixeira, University of Central Florida - EUA.

Europe/Lisbon
Room P4.35, Mathematics Building — Online

Ram Band

Ram Band, Israel Institute of Technology
The dry ten Martini problem for Sturmian Schrödinger operators

"Are all gaps there?", asked Mark Kac in 1981 during a talk at the AMS annual meeting, and offered ten Martinis for the answer.

This led Barry Simon to coin the names the Ten Martini Problem (TMP) and the Dry Ten Martini Problem for two related problems concerning the Almost-Mathieu operator.

The TMP is about showing that the spectrum of the Almost-Mathieu operator is a Cantor set. The Dry TMP is about the values that the integrated density of states (IDS) attains at the spectral gaps. The gap labelling theorem predicts the possible set of values which the IDS may attain at the spectral gaps. The Dry TMP is whether or not all these values are attained, or equivalently, "are all gaps there?". The TMP was fully solved by Artur Avila and Svetlana Jitomirskaya in 2005.

Artur Avila, Jiangong You and Qi Zhou posted this year a preprint with the solution of the Dry TMP for the non-critical case (the coupling constant of the potential differs from one).

This talk is about the Dry TMP for Sturmian Schrödinger operators. These are one-dimensional Schrödinger operators with aperiodic potentials which are Sturmian sequences. The potential is determined in terms of two parameters: the frequency and the potential strength (a.k.a coupling constant). As for the Almost-Mathieu operator the Dry TMP is whether all the possible spectral gaps are there for all irrational frequencies and all values of the coupling constant. For large values of the coupling constant, the Sturmian Dry TMP was solved by Laurent Raymond in 1995. In 2016, David Damanik, Anton Gorodetski and William Yessen provided a solution if the frequency is the golden mean and for all couplings.

We present our current work, jointly with Siegfried Beckus and Raphael Loewy, where we solve the Sturmian Dry TMP for all irrational frequencies and all couplings.

Europe/Lisbon
Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa — Online

Hyeong-Ohk Bae, Ajou University, Republic of Korea
Interaction of particles and an incompressible fluid

We present a new coupled kinetic-fluid model for the interactions between Cucker-Smale (C-S) flocking particles and incompressible fluid on the periodic spatial domain $\mathbb T^d$ and in an infinite channel.

Our coupled system consists of the kinetic Cucker-Smale equation and the incompressible Navier-Stokes equations, and these two systems are coupled through the drag force. For the proposed model, we provide a global existence of weak solutions and a priori time-asymptotic exponential flocking estimates for any smooth flow, when the kinematic viscosity of the fluid is sufficiently large. The velocity of an individual C-S particle and fluid velocity tend to the averaged time-dependent particle velocities exponentially fast.