Mon, 26 Apr 2021

16:00 - 17:00
Virtual

On the minimization of convex, variational integrals of linear growth

Lisa Beck
(University of Augsburg)
Abstract

We study the minimization of functionals of the form $$ u  \mapsto \int_\Omega  f(\nabla u) \, dx $$

with a convex integrand $f$ of linear growth (such as the area integrand), among all functions in the Sobolev space W$^{1,1}$ with prescribed boundary values. Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and might in fact fail, as it is well-known already for the non-parametric minimal surface problem. In such cases, the functional is extended suitably to the space BV of functions of bounded variation via relaxation, and for the relaxed functional one can in turn guarantee the existence of minimizers. However, in contrast to the original minimization problem, these BV minimizers might in principle have interior jump discontinuities or not attain the prescribed boundary values.

After a short introduction to the problem I want to focus on the question of regularity of BV minimizers. In past years, Sobolev regularity was established provided that the lack of ellipticity -- which is always inherent for such linear growth integrands -- is mild, while, in general, only some structure results seems to be within reach. In this regard, I will review several results which were obtained in cooperation with Miroslav Bulíček (Prague), Franz Gmeineder (Bonn), Erika Maringová (Vienna), and Thomas Schmidt (Hamburg).

Thu, 07 Nov 2019

12:00 - 13:00
L4

A new Federer-type characterization of sets of finite perimeter

Panu Lahti
(University of Augsburg)
Abstract

Federer’s characterization, which is a central result in the theory of functions of bounded variation, states that a set is of finite perimeter if and only if n−1-dimensional Hausdorff measure of the set's measure-theoretic boundary is finite. The measure-theoretic boundary consists of those points where both the set and its complement have positive upper density. I show that the characterization remains true if the measure-theoretic boundary is replaced by a smaller boundary consisting of those points where the lower densities of both the set and its complement are at least a given positive constant.

Mon, 11 Feb 2019
15:45
L6

Local flexibility for open partial differential relations

Bernhard Hanke
(University of Augsburg)
Abstract

In his famous book on partial differential relations Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental assertion and sketch a proof. 

In the sequel we will apply this to extend local deformations of closed $G_2$ structures, and to construct 
$C^{1,1}$-Riemannian metrics which are positively curved "almost everywhere" on arbitrary manifolds. 

This is joint work with Christian Bär (Potsdam).

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