Singularities of energy minimizing harmonic mappings from the ball to the sphere
Dienstag, 18.10.16, 16:15-17:15, Raum 404, Eckerstr. 1
Minimizing harmonic maps (i.e. minimizers of the Dirichlet integral) with prescribed boundary conditions from the ball to the sphere may have singularities. For some boundary data it is known that all minimizers of the energy have singularities and the energy is strictly smaller than the infimum of the energy among the continuous extensions (the so called Lavrentiev gap phenomenon occurs). We prove that the Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data. This is joint work with P. Strzelecki.
Sectional and intermediate Ricci curvature bounds via optimal transport
Dienstag, 25.10.16, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk we present an optimal transport characterization of lower sectional curvature bounds for smooth Riemannian manifolds. More generally, we characterize lower bounds for the p-Ricci tensor in terms of convexity of the relative Reny entropy on Wasserstein space with respect to the p-dimensional Hausdorff measure. The p-Ricci tensor corresponds to taking the trace of the Riemannian curvature tensor on p-dimensional planes. This is a joint work with Andrea Mondino.
Steklov-eigenvalue bounds and minimal surface I
Dienstag, 8.11.16, 16:15-17:15, Raum 404, Eckerstr. 1
We present results by Frasier and Schoen written down in their paper "Sharp eigenvalue bounds and minimal surfaces in the ball". In the first talk we discuss properties of the first Steklov eigenvalue and lay the requirements to prove in the second talk that under all annulus' the critical one gives the maximal Steklov eigenvalue.
Steklov-eigenvalue bounds and minimal surface II
Dienstag, 22.11.16, 16:15-17:15, Raum 404, Eckerstr. 1
Introduction to first variation of varifolds
Dienstag, 29.11.16, 16:15-17:15, Raum 404, Eckerstr. 1
Introduction to general varifolds
Dienstag, 20.12.16, 16:15-17:15, Raum 404, Eckerstr. 1
Monotonicity Formula for Varifolds
Dienstag, 10.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
Connection between \(p\)-harmonic functions and the inverse mean curvature flow
Dienstag, 17.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk we will present a generalisation of Roger Mosers method for an alternative proof for the existence of solutions to the IMCF in the case of a complete and closed Riemannian manifold with bounded curvature. In the course of doing so we will also discuss why certain properties of \(p\)-harmonic functions, such as the Hölder-continuity of the gradient, will be preserved and give an outlook on further proceedings in the field.
Regularity for some elliptic equations with orthotropic structure
Dienstag, 24.1.17, 16:15-17:15, Raum 404, Eckerstr. 1
We discuss a variant of the p-Laplacian operator, which arises as the first variation of a suitable Dirichlet integral. The corresponding elliptic equation is much more degenerate/singular than that for the standard p-Laplacian operator and higher regularity of solutions is a difficult issue.\nWe will present some regularity results for the gradient of solutions (differentiability, boundedness and continuity), mainly for the two dimensional case. We will also briefly address the case of nonstandard growth conditions and the higher dimensional case.\nThe results presented are contained in some papers in collaboration with Pierre Bousquet (Toulouse), Guillaume Carlier (Paris Dauphine), Vesa Julin (Jyvaskyla), Chiara Leone (Napoli), Giovanni Pisante\n(Caserta) and Anna Verde (Napoli).
A two-phase free boundary problem for the fractional Laplacian
Dienstag, 24.1.17, 17:15-18:15, Raum 404, Eckerstr. 1
In this talk, I will discuss a non-local free boundary problem of two-phase type, related to the fractional Laplacian. In particular, I will discuss the optimal regularity and the separation of phases. It turns out that certain non-local problems differ from their local siblings, in the sense that the two phases can never meet. This is joint work with Mark Allen and Arshak Petrosyan.\n\n\n\n