**Nonlinear PDE Days Frankfurt - Gießen - Karlsruhe - Köln**

**Local and Nonlocal Problems in Geometry and Mathematical Physics**

**Goethe-Universität Frankfurt, August 22 - 23, 2018**

**Speakers**

**Dorin Bucur (Université de Savoie)**

**Michel Chipot (Universität Zürich)**

**Andrea Cianchi (Università degli Studi di Firenze)**

**Eleonora Cinti (Università di Bologna) **

**Mouhamed Moustapha Fall (AIMS Senegal, Mbour) **

**Enno Lenzmann (Universität Basel)**

**Organizers**

Thomas Bartsch (Universität Gießen)

Dirk Hundertmark (KIT Karlsruhe)

Bernd Kawohl (Universität zu Köln)

Tobias Lamm (KIT Karlsruhe)

Mohameden Ahmedou (Universität Gießen)

Michael Plum (KIT Karlsruhe)

Wolfgang Reichel (KIT Karlsruhe)

Guido Sweers (Universität zu Köln)

Tobias Weth (Universität Frankfurt)

**Schedule**

**Wednesday, August 22, 2018** **2:00 pm Dorin Bucur**

**Optimal partition problems and the honeycomb conjecture**

**Abstract:** In 2005-2007 Burdzy, Caffarelli and Lin, Van den Berg conjectured in different contexts that the sum (or the maximum) of the first eigenvalues of the Dirichlet-Laplacian associated to arbitrary cells partitioning a given domain of the plane, is asymptomatically minimal on honeycomb structures, when the number of cells goes to infinity. I will discuss the history of this conjecture, giving the arguments of Toth and Hales on the classical honeycomb problem, and I will prove the conjecture (of the maximum) for the Robin-Laplacian eigenvalues.

**3:00 pm Eleonora Cinti**

**Some recent results in the study of nonlocal minimal surfaces**

**Abstract:** We present some recent results concerning the classiﬁcation of nonlocal minimal surfaces. More precisely, we prove a quantitative ﬂatness result in dimension n = 2 and some energy and BV estimates for stable sets for the fractional s-perimeter. This estimates are also a crucial ingredient in the proof of a classiﬁcation result in dimension n = 3 for s close to 1. These results are contained in two works in collaboration with X. Cabré, J. Serra and E. Valdinoci.

4:00 Coffee Break

**4:45 pm Michel Chipot **

**On some stationary Navier-Stokes type problems**

**Abstract:** The goal of this talk is to present a simple proof of existence of solution to the stationary Navier-Stokes problem using a singular perturbation technique and to address some nonlocal issues related to it.

**Thursday, August 23, 2018 **

**9:00 am Andrea Cianchi**

**Nonlinear elliptic equations under minimal data and domain regularity.**

**Abstract:** I will discuss a few aspects of the regularity of solutions to boundary value problems for nonlinear elliptic equations and systems of p-Laplacian type. In particular, second-order regularity properties of solutions, and the boundedness of their gradient will be focused. The results to be presented are optimal, in a sense, as far as the regularity of the right-hand sides of the equations and the boundary of the underlying domains are concerned. The talk is based on joint researches with V.Maz'ya.

10:00 Coffee Break

**10:30 am Enno Lenzmann **

**Rearrangement in Fourier Space and Symmetry Results for higher-order PDE**

**Abstract:** In this talk, I discuss a new method to prove general symmetry results for variational problems in R^n involving (pseudo)-differential operators of arbitrary order. In particular, we can deal with problems, where the classical Polya-Szegö principle fails to be applicable. The method is based on symmetric-decreasing rearrangements in Fourier space. We obtain a class of sharp rearrangement inequalities. As some applications, we prove radial symmetry of optimizers for Gagliardo-Nirenberg inequalities in L^2 with arbitrary differential order and some general Adams-Moser-Trudinger type in equalities in R^n. This is joint work with Jérémy Sok.

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11:30 am Mouhamed Moustapha Fall

**Critical points of the torsional rigidity energy and related problems**

**Abstract:** In 1856, Saint-Venant conjectured that among beams with constant simply connected cross sections, the one with circular cross section maximizes the torsional rigidity energy. Starting from the proof of of Pòlya in 1948, positive answers to this conjectured have been given by several people in the last decades, using Riemann mapping theorem or symmetrization techniques. The question of maximizing torsional rigidity leads in general to the study of partial differential equations with "constant" nonzero Dirichlet, Neuman and Source. By means of the Alexandrov's moving plane argument, Serrin classified, in 1971, all stationary "bounded" regular domains which are stationary sets ( or critical points) to the torsional rigidity energy. In this lecture, we will also provide recent results on "unbounded" sets that are stationary sets to the torsional rigidity energy together with connection to Constant (Nonlocal) Mean Curvature surfaces.

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**Location**

Institut für Mathematik

Robert-Mayer-Straße 10 (Corner Gräfstraße/Robert-Mayer-Straße)

60325 Frankfurt am Main

Lecture Room 711gr

#### Map

**Directions from Frankfurt HBF (main railway station)**

Take the metro train U4 to Bockenheimer Warte (two stations, direction Bockenheimer Warte). When you exit the train at Bockenheimer Warte, take the stairs reverse to the direction of travel, turn right and take stairs again to get to the street level.

Then follow the street in the direction of your exit (called Senckenberganlage). The next street to your right will be the Robert-Mayer-Strasse. You turn right and follow the Robert-Mayer-Strasse until number 10. The lecture room 711 gr. is on 7th floor, the workshop office (Mrs Habash) is room 802 located on 8th floor.

If the metro trip is not covered by your train ticket already, you can buy a ticket called "Kurzstrecke".

**Directions from Frankfurt International Airport**

Go to the local railway station for regional and local trains ("S-Bahn"), located at Terminal 1, Level 1. There you will find a ticket machine where you buy a ticket to Frankfurt City. Then go down to platform 1 and take one of the lines S8 or S9 (direction Frankfurt Hbf., Offenbach or Hanau) to Frankfurt HBF, the main railway station. Then proceed as described above, taking the metro train U4 to Bockenheimer Warte.

Please note that you have to buy the ticket before you enter the train; you cannot buy it inside the train.

**Contact **

To register for participation, please mail to Jacqueline Habash

Registration is free.

We will be happy to help you with your hotel reservation.