GEOMETRIC PDE's DAY
10:30-11:30: Niels Martin Molles (University of Copenhagen)
Rigidity of the grim reaper cylinder as a collapsed self-translating soliton
Mean curvature flow self-translating solitons are minimal hypersurfaces for a certain incomplete conformal background metric, and are among the possible singularity models for the flow. In the collapsed case, they are confined to slabs in space. The simplest non-trivial such example, the grim reaper curve \(\Gamma\) in \(\mathbb{R}^2\), has been known since 1956, as an explicit ODE-solution, which also easily gave its uniqueness.
We consider here the case of surfaces, where the rigidity result for \(\Gamma\times\mathbb{R}\) that we'll show is:
The grim reaper cylinder is the unique (up to rigid motions) finite entropy unit speed self-translating surface which has width equal to \(\pi\) and is bounded from below. (Joint w Impera & Rimoldi.)
Time permitting, we'll also discuss recent uniqueness results in the collapsed simply-connected low entropy case (joint w Gama & Martín), using Morse theory and nodal set techniques, which extend Chini's classification.
12:00-13:00: David Ruiz (University of Granada)
A Schiffer-type problem for annular domains
If a nonconstant Neumann eigenfunction \(u\) of the Laplacian on a smooth bounded domain in \(\mathbb{R}^2\) is constant on the boundary, then the domain is a disk.
In this talk we will consider a version of such question for domains with disconnected boundary. Specifically, we consider Neumann eigenfunctions that are locally constant on the boundary, and we wonder if the domain has to be necessarily a disk or an annulus.
We will show that the answer to the above question is negative. Indeed, there are nonradial Neumann eigenfunctions which are locally constant on the boundary of the domain. The proof uses a local bifurcation argument together with a reformulation of the problem by Fall, Minlend and Weth that avoids a problem of loss of derivatives. This is joint work with A. Enciso, A. J. Fernández and P. Sicbaldi.
16:00-17:00: Eddygledson Souza Gama (Universidade Federal de Pernambuco)
New examples of translating solitons
In this talking we are going to discuss about new examples of translating solitons with entropy five and six.
