Regularity theory for elliptic and parabolic systems and problems in continuum mechanics

Telč, Czech Republic, 6.–9. 5. 2020

Program

To be announced.

Invited lectures

Michael Bildhauer

Some analytical results for image restoration methods of linear growth type

Different linear growth methods for the restoration of images are discussed. In the first part we concentrate on several variational approaches, which in the case of appropriate ellipticity conditions lead to regular solutions. The so called Edge-Enhancing Anisotropic Diffusion used for image compression is studied in the second part of the talk. The underlying integrodifferential operator under consideration is based on the so called Charbonnier diffusivity w.r.t. a Gaussian kernel which operates across the edge direction. Here we prove the existence of solutions via fixed point arguments and discuss apriori estimates of suitable iterated sequences. The results were mainly obtained as joint work with Marcelo Cárdenas, Martin Fuchs and Joachim Weickert.

Gilles A. Francfort

Regularity and rigidity of the stress field in Von Mises plasticity

In this joint work with J. F. Babadjian, inspired from prior work with A. Giacomini and J. J. Marigo, we start an investigation of spatial hyperbolicity in Von Mises elasto-plasticity, the ultimate goal being an adjudication of the uniqueness of the plastic strain. After discussing a specific example where uniqueness, or lack thereof, can be established, I will present partial results, focussing on a 2d simplified model.

Lenka Slavíková

A sharp boundedness criterion for bilinear Fourier multipliers

A linear Fourier multiplier operator associated with an arbitrary bounded function is always $$L^2$$-bounded. The related question of the $$L^2\times L^2 \to L^1$$ boundedness of bilinear Fourier multiplier operators is however much less straightforward; in particular, there are bounded functions for which the associated bilinear operator is unbounded from $$L^2 \times L^2$$ to $$L^1$$. In this talk we present a sharp criterion for the $$L^2 \times L^2 \to L^1$$ boundedness of bilinear Fourier multipliers. This is a joint work with Loukas Grafakos and Danqing He.