We introduce two different definitions for mappings of bounded variation between a metric measure space and a metric space. We will proof that whenever the domain is doubling and supports a 1-Poincaré inequality and the target space is a Banach space then both definitions of bounded variation coincide, but we will give examples that this is not the case when the target is a general metric space, even when the domain has good geometry. The talk is based on a joint work with Josh Kline and Nages Shanmugalingam.
In this talk we will address different variants of the classical Rademacher theorem in the setting of metric measure spaces. Some of he fundamental results by Kirchheim and Cheeger in this direction will be briefly reviewed. Furthermore, we will consider the notion of metric differentiability defined in terms of Lipschitz charts of the space, and its connections with rectifiability properties of the underlying space. The talk is based on joint work with I. Caamaño, E. Durand-Cartagena, A. Prieto and E. Soultanis.
The classical Rademacher's theorem on the a.e. differentiability of Lipschitz functions on Euclidean space has a famous generalization (due to Cheeger) to metric measure spaces (mms) supporting a Poincare inequality. This result has inspired a large body of research on the differentiability of Lipschitz functions on mms, in which the role of curve fragments as a substitute for directional derivatives has become apparent. In this talk I discuss the connection between differentiability of Lipschitz functions and metric Sobolev spaces. In particular I'll describe how, using techniques from metric Sobolev spaces, one can establish some weak (i.e. directional) differentiability of Lipschitz functions with essentially no assumptions on the mms.
Relativistic Elasticity in Minkowski flat spacetime, or more generally in Lorentzian manifolds, has been the subject of recent studies in mathematical physics, as well as in physics (e.g. studies by Kolev, Desmorat and Hudelist). During the talk we define the Minkowski flat spacetime, as well as, some of its generalisations. Furthermore, we will introduce the concept of relativistic elasticity and formulate some of the associated results. Lastly, we will reformulate some of the related tensors into the language of matrices.
Schedule of the seminar.
After discussing a suitable definition of the variation energy for mappings between a metric measure space and a metric space we will introduce a notion of jump point equivalent to that of a point of non-approximate continuity. Thanks to the metric nature of this definition and the theory of sets of finite perimeter we will prove, for a map of bounded variation, the uniform finitenes of jump values (i.e., values attained at a positive density) for codimension 1 a.e. jump point whenever the source space is doubling and supports a 1-PI and the target space is complete and compact.
Some 15 years ago, Heiko von der Mosel and the speaker have analyzed several nonlocal energies of non-smooth sets S, defined as multiple integrals of various geometrically defined quantities, depending on two or more points of S. One of them was the so-called tangent-point energy, equal to the double integral over S, with respect to the Hausdorff measure, of a function depending on two points x,y of S: (a negative power of) the radius R of a sphere which is tangent to S at the first point, x, and passes through the second point, y. For this energy, optimal regularity results (for sets S having finite energy) are known; there is also a full characterization (in terms of fractional Sobolev spaces) of those S that have finite energy, due to Simon Blatt.
I will present a survey of those results and indicate new proofs of optimal regularity and of Blatt's characterization which are simpler than those previously known. They are both based on a new lemma which holds for graphs with finite tangent-point energy and is reminiscent of the celebrated Piotr Hajlasz's characterization of Sobolev spaces in terms of upper gradients; the consequences and possible analogies still remain to be fully explored.
Gromov proved in 1996 that there is no Holder continuous $C^{0,\alpha}$ embedding of $\mathbb{R}^3$ into the Heisenberg group $\mathbb{H}^1$ when $\alpha>\frac{2}{3}$. More generally, there is no $C^{0,\alpha}$ embedding of $\mathbb{R}^m$ to $\mathbb{H}^n$ when $m\geq n+1$ and $\alpha>\frac{n+1}{n+2}$. The original proof was difficult. I will discuss a new elementary and geometric proof of a slightly stronger result discovered by Hajłasz and Schikorra. The proof is based on the notion of distributional Jacobian.