PDEs unplugged - mathematics only with chalk and blackboard - no technical gimmicks and gadgets

In 2015 a unique new format called unplugged in PDE's was initiated by our colleagues Massimo Grossi and Angela Pistoia from La Sapienza University, Rome.

A truly wonderful event with exclusively blackboard talks (and mobile phones, laptops switched off). We were so much delighted by the format that we now repeat this format on June 13/14 2017 as PDEs_unplugged @ Karlsruhe. Everyone feel free to copy this format.

## Schedule

**Tuesday, June 13**

14:00-14:50 | Catherine Bandle | Semilinear elliptic problems with a Hardy potential |

14:55-15:45 | Nalini Anantharaman | Dispersion and controllability for linear Schrödinger equations on compact Riemannian manifolds |

Coffee break | ||

16:15-17:05 | Jacopo Bellazzini | Long time dynamics for semirelativistic NLS and Half Wave in arbitrary dimension |

17:10-18:00 | Jean-Baptiste Casteras | Stability of ground state and renormalized solutions to a fourth order Schrödinger equation |

**Wednesday, June 14**

9:00-9:50 | Lisa Beck | Lipschitz regularity and non-uniform ellipticity |

9:55-10:45 | Alberto Saldana | Existence and qualitative properties of elliptic Hamiltonian systems with Neumann boundary conditions |

Coffee break | ||

11:15-12:05 | Massimo Grossi | Radial nodal solutions for Moser-Trudinger problems |

12:10-13:00 | Michel Willem | On some weakly coercive quasilinear problems with a forcing term |

**Campus map**

## Speakers/Abstracts

**Catherine Bandle (Universität Basel, Schweiz)**

*Semilinear elliptic problems with a Hardy potential*

We discuss semilinear elliptic equations in a bounded domain, with a power nonlinearity and a Hardy potential which is singular at the boundary. The boundary behavior of the positive solutions is determined either by the nonlinearity or by the solutions of the linear problem. The presence of the Hardy potential forces a solution to vanish or to become infinite a the boundary. The first results of this type were obtained in collaboration with V. Moroz and W. Reichel.

**Nalini Anantharaman (Université de Strasbourg, France)**

*Dispersion and controllability for linear Schrödinger equations on compact Riemannian manifolds*

I will review several results related to the controllability and dispersive properties of the linear Schrödinger equation on compact Riemannian manifolds, putting the emphasis on the role of the geometry. Our main goal is to obtain observability inequalities in situations where the geometric control condition is not fulfilled. I will discuss, in particular, the case of negatively curved manifolds, where the classical propagation of rays is chaotic, and of flat tori and of the 2-dimensional disk, where the classical propagation is completely integrable. Joint work with Gabriel Rivière, Fabricio Maciá, Matthieu Léautaud, Clotilde Fermanian.

**Jacopo Bellazzini (Università degli Studi di Sassari, Italia)**

*Long time dynamics for semirelativistic NLS and Half Wave in arbitrary dimension*

Aim of the talk is to discuss the dynamic of semirelativistc NLS (sNLS) and half wave (HW) in arbitrary dimension. Concerning sNLS we show the existence and stability of ground states in the mass supercritical regime. This is in sharp contrast with the instability of ground states for the corresponding HW by showing an inflation of norms phenomenon. Joint work with Vladimir Georgiev and Nicola Visciglia.

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**Jean-Baptiste Casteras (Université libre de Bruxelles, Belgique)**

*Stability of ground state and renormalized solutions to a fourth order Schrödinger equation*

In this talk, we will be interested in standing wave solutions to a fourth order nonlinear Schrödinger equation having second and fourth order dispersion terms. This kind of equation naturally appears in nonlinear optics. In a first time, we will establish the existence of ground-state and renormalized solutions. We will then be interested in their qualitative properties, in particular their stability. Joint works with Denis Bonheure, Ederson Moreira Dos Santos, Tianxiang Gou, Louis Jeanjean and Robson Nascimento.

**Lisa Beck (Universität Augsburg)**

*Lipschitz regularity and non-uniform ellipticity*

We discuss some sharp gradient regularity criteria for minimizers of convex variational integrals of the form As the main feature, we allow the ellipticity ratio (of largest to smallest eigenvalue of ) to be unbounded, which means that the associated Euler-Lagrange equation may be non-uniformly elliptic. This phenomenon happens in particular for functionals exhibiting non-standard -growth. For this by now classical example we explain how to establish, under the minimal regularity assumption that belongs to the Lorentz space , local Lipschitz continuity estimates, which extend the classical ones for harmonic and -harmonic functions. The proof is based on nonlinear potential theoretic arguments, which combine Caccioppoli-type inequalities involving intrinsic quantities with an iteration developed by De Giorgi, Kilpeläinen and Malý. This strategy of proof extends to more general growth assumptions, which cover in particular the one of anisotropic growth, but also of fast growth of exponential type investigated in recent years. The results presented in this talk are part of a joined project with Giuseppe Mingione (Parma).

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**Alberto Saldana (KIT)**

*Existence and qualitative properties of elliptic Hamiltonian systems with Neumann boundary conditions*

Systems of equations where the components have nonlinear interactions are present in many mathematical models coming from physics, biology, and chemistry. However, the nature of nonlinear interactions can be very complex and it is still poorly understood. One of the simplest models one can consider to study this kind of structure are the so-called elliptic Hamiltonian systems. In this talk, we consider an elliptic Hamiltonian system with power nonlinearities and Neumann boundary conditions. We explain the existence theory based mostly on the dual method, and derive some qualitative properties such as: unique continuation principles, axial symmetry of global minimizers, monotonicity of radial minimizers, and symmetry-breaking phenomena. As a particular case, our results also describe the properties of solutions to the scalar equation. This is joint work with Hugo Tavares.

**Massimo Grossi (Sapienza - Università di Roma, Italia)**

*Radial nodal solutions for Moser-Trudinger problems*

In 1992 Adimurthi and Yadava introduced some nonlinearities for Moser-Trudinger type problems in the ball which are sharp with respect to the existence of solutions. We study the asymptotic behaviour of such solutions. This is a joint paper with Daisuke Naimen (Muroran, Japan).

**Michel Willem (Université catholique de Louvain, Belgique)**

*On some weakly coercive quasilinear problems with a forcing term*

We consider the forced problem where the functional is positive definite on but not necessarily coercive and where the distribution is such that

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