WADE: from Lisbon to the world

The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSDCMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.

All seminars will take place in the Zoom platform. The links for the planned seminars can be found here. In order to get the password to access the seminars, please subscribe the announcements in the registration menu, or contact one of the organizers.

 

02/06/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online *

Maria Colombo (École Polytechnique Fédérale de Lausanne)

Nonunique characteristic curves of Sobolev vector fields

Given a vector field in Image removed., the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth; this, in turn, translates in existence and uniqueness results for the transport equation. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollaryof the uniqueness of the trajectory of the ODE for a.e. initial datum. In this talk we give an overview of the topic and we provide a negative answer to this question. To show this result we exploit the connection with the transport equation, based on Ambrosio’s superposition principle, and a new ill-posedness result for positive solutions of the continuity equation.

 *to get access to the password, please register on the website or contact one of the organizers.

 

26/05/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online *

 Luis Vega (Basque Center for Applied Mathematics)

The Vortex Filament Equation, the Talbot effect, and non-circular jets 

We will propose the vortex filament equation as a possible toy model for turbulence, in particular because of its striking similarity to the dynamics of non-circular jets. This similarity implies the existence of some type of Talbot effect due to the interaction of non-linear waves that propagate along the filament. Another consequence of this interaction is the existence of a new class of multi-fractal sets that can be seen as a generalization of the graph of Riemann’s non-differentiable function. Theoretical and numerical arguments about the transfer of energy will be also given. This a joint work with V. Banica and F. de la Hoz.

 *to get access to the password, please register on the website or contact one of the organizers.

 

19/05/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online *

Bernold Fiedler (Institute of Mathematics, Freie Universität Berlin)

Sturm meanders: global attractors, Temperley-Lieb algebras, and black holes

 Fusco and Rocha studied Neumann boundary value problems for scalar ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutations. These permutations relate, on the one hand, to a special class of meandering curves as introduced by Arnol’d in a singularity theory context. On the other hand, they became central in the study of global attractors of nonlinear parabolic partial differential equations of Sturm type.

We discuss relations of Sturm meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.

We report on joint work with Pablo Castañeda, Juliette Hell, Carlos Rocha, and Brian Smith. See also http://dynamics.mi.fu-berlin.de/

For further material we recommend the beautifully illustrated book “Meanders” by Anna Karnauhova, de Gruyter 2017.

 

*to get access to the password, please register on the website or contact one of the organizers.

 

12/05/2020, Tuesday, 16:30–17:30 Europe/Lisbon — Online *

Diogo Oliveira e Silva (University of Birmingham)

Global maximizers for spherical restriction

Image removed.

 *to get access to the password, please register on the website or contact one of the organizers.

 

05/05/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online *

Filippo Santambrogio (Université Claude Bernard - Lyon 1)

Optimal transport methods for the regularity of 2D functions of least gradient

The least gradient problem (minimizing the BV norm with given boundary data), motivated by both image processing applications and connections with minimal surfaces, is known to be equivalent, in the plane, to the Beckmann minimal-flow problem (an alternative formulation of the L1 Monge-Kantorovich optimal transport problem) with source and target measures located on the boundary of the domain. Hence, Sobolev regularity of functions of least gradient is equivalent in this setting to Lp bounds on the solution of the Beckmann problem (i.e. on what is called the transport density) and can be attacked with techniques which are now standard in optimal transport. From the transport point of view, the novelty of the estimates that I will present, coming from a joint paper with S. Dweik, lies in the fact they are obtained for transport between measures which are concentrated on the boundary. From the BV point of view, a new result is the W1,p regularity of the least gradient function whenever the boundary datum is W1,p  as a 1D function: moreover, the optimal transport framework is strong enough to deal with arbitrary strictly convex norms instead of the Euclidean one with almost no effort.

 

*to get access to the password, please register on the website or contact one of the organizers.

 

The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.

 

 

28/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online 

Michael Goldman (Laboratoire Jacques-Louis Lions and Université Paris 7) 

On an old conjecture of Almgren

In this talk I will give an overview on the few results available on the conjecture of Almgren regarding the convexity of drops subject to the action of an external potential. In particular I will present recent progress in this direction obtained with G. De Philippis on their connectedness. Together with an older result of McCann, this answers positively the conjecture in dimension two. The proof is inspired by the two-point function technique introduced by B. Andrews and is reminiscent of the doubling of variables trick in the context of viscosity solutions.

 

 

21/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online

Matthias Röger, Technische Universität Dortmund

An obstacle type problem as a limit of a model for cell polarization

We consider the polarization of a cell in response to an outer signal. The mathematical model consists of a diffusion equation in the inner volume coupled to a reaction diffusion system on the cell membrane. In a certain asymptotic limit we rigorously prove the convergence towards a generalized obstacle problem. In term of this limit system we derive conditions for the onset of polarization. The results will be mainly presented for the stationary case, we will briefly discuss some extensions to the time-dependent case.

(This is joint work with Barbara Niethammer, Juan Velazquez, and Anna Logioti)

 

14/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon — Online

Dorin Bucur, Université de Savoie.

Boundary behaviour of Robin problems and isoperimetric spectral inequalities.

Consider the Poisson equation with Robin boundary conditions in a (nonsmooth) domain with a bounded, nonnegative right hand side. Given a point on the boundary, the question is whether the solution has a strictly positive lower limit at this point. If the domain is smooth the answer is positive as a consequence of the Hopf maximum principle. If the domain is not smooth, the answer may be positive or negative, depending on the geometry of the domain around the point. This question was raised in a probabilistic context by Bass, Burdzy and Chen in 2008, when they obtained results for Lipschitz sets and cuspidal domains.

Our motivation is related to the fact that positive answers to the question above, together with a control of the infimum of the boundary values, lead to sharp quantitative forms of isoperimetric inequalities of spectral type for the Robin Laplacian.

In this talk, I will make the point on recent advances on isoperimetric inequalities involving the Robin Laplacian and I will show how the boundary behaviour pops up in the quantitative form of the inequalities. If times remains, I will present a variational approach to deal with the boundary behaviour for general elliptic operators and various geometric contexts.

The results were obtained together with A. Giacomini and M. Nahon.

 

07/04/2020, Tuesday, 16:00–17:00 Europe/Lisbon —  Online

Susanna Terracini, Università di Torino.
Pattern Formation Through Spatial Segregation.

Susanna Terracini

Video available here.