Seminar on partial differential equations and applied topics

The usual meeting time is Tuesdays at 13:30 in room 109c, Robert-Mayer-Straße 10.

Important note: At the moment, the building can only be accessed if you are vaccinated, cured, or have a negative test which is not older than 24 hours.

List of talks:

2022 April 26th, Antonio Greco (University of Cagliari)

Topic: Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains
Abstract: We consider semilinear PDEs in a sector-like domain. Using cylindrical coordinates (r,θ,z), we investigate the shape of solutions whose derivative in θ vanishes at the boundary. Assuming that the non-linearity f(r,z,u) is convex in u and does not depend on θ, we prove that any solution with Morse index less than two must be either independent of θ or strictly monotone with respect to θ. In the special case of a planar domain, the result holds in a circular sector as well as in an annular, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case when its opening is larger than π (joint work with Francesca Gladiali on arXiv:2201.11988).

2022 January 25th, Sarah Kistner (Uni Frankfurt)

Topic: The logarithmic Laplacian and the fractional Poisson problem
Abstract: We study the differentiability of the solition map to the fractional Poisson problem and introduce the logarithmic Laplacian. We pay special attention to the fractional Torsionproblem and some function which appears in this setting. Finally we discuss two possibilities to derive the explicit expression of this function in some open, bounded sets and show that this function is quasiconvex if our set is convex.

2021 December 7th, Joel Kübler (Uni Frankfurt)

Topic: Rotating Solutions of Nonlinear Klein-Gordon Equations with 'high speed'
Abstract: We study solutions of nonlinear Klein-Gordon equations with power-type nonlinearities, whose time-dependence is characterized by a rotation. We will focus on the case with 'high velocity', which leads to an elliptic-hyperbolic equation.

Based on a detailed study of the spectrum of the associated mixed-type operator, we will use dual variational methods to find nontrivial solutions.

2021 November 30th, Ignace Aristide Minlend (Uni Frankfurt)

Topic: Foliation of an Asymptotically flat end by critical capacitors.
Abstract: We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory.

These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary
value problem involving the Laplace-Beltrami operator. In a key step we must solve a nonlinear equation driven by the Dirichlet-to-Neumann operator.

2021 November 23th, Omar Cabrera Chávez (Uni Frankfurt)

Topic: Partial ground states for the logarithmic Choquard equation in R^2
Abstract: see here.

2021 November 9th, Pierre Aime Feulefack (Uni Frankfurt, AIMS Senegal)

Topic: On nonlocal operators of small order
Abstract: In this talk, we present some nonlocal operators with differential order close to zero. In particular, we will focus on the proof of the interior regularity of weak solutions to the associated linear elliptic integro-differential equations. Our approach exploits the variational structure of the problem combined with an intermediate estimate in Nikol'skii spaces.

2021 November 2nd, Remi Yvant Temgoua (Uni Frankfurt, AIMS Senegal)

Topic: Existence results for regional fractional Laplacian: minimizers for Sobolev constant
Abstract: In this talk, we study the best constant appearing in Sobolev inequality in domains. Our main focus will be to analyze the attainability of both this constant and its radial counterpart via the method of missing mass due to Lieb. We believe this produces new insights into the study of minimization problems involving regional fractional Laplacian.

2021 October 26th, Sidy Moctar Djitte (Uni Frankfurt, AIMS Senegal)

Topic: Maximization of the second eigenvalue of the fractional Laplacian
Abstract: (see also here) In the first part of the talk, we discuss symmetry properties of second eigenfunctions of the annulus. We prove that for an annulus with a sufficiently large hole, a second eigenfunction cannot be radial. This gives a partial answer to a conjecture by Bunuelos and Kulczycki on the shape of second eigenfunctions of annuli. In the second parts we consider the maximization of the second eigenvalue in simply connected domains bounded by two spheres and prove that the maximum is attained by concentric spheres.

The talk is based on two papers: A fractional Hadamard formula and Applications joint with M.M. Fall and Tobias Weth; Symmetry of odd solutions to equations in involving the fractional Laplacian joint with Sven Jarohs.

2021 October 19th, Sven Jarohs (Uni Frankfurt)

Topic: The fractional Bernoulli problem
Abstract: In this talk I present some recent results of a joint work with Paolo Salani and Tadeusz Kulczycki concerning the fractional Bernoulli problem, where I will focus on the so called spectral fractional Laplacian. After a short introduction, I will explain how to construct a solution to this problem using the Beurling method on the extended problem. Moreover, we discuss geometric properties of the solution and, if there is time, give some remarks on the problem with the usual fractional Laplacian.