Perimeter functionals with measure datum

Abstract: The talk is concerned with perimeter functionals $\mathscr{P}_\mu$ given by \[ \mathscr{P}_\mu[A]:=\mathrm{P}(A)-\mu(A^+) \] on sets $A\subset{\mathbb R}^n$ of finite volume and finite perimeter $\mathrm{P}(A)$, where the fixed non-negative Radon measure $\mu$ may be singular and is (necessarily) evaluated on a suitable closure $A^+$ of $A$. It will be explained that semicontinuity and existence results for $\mathscr{P}_\mu$ crucially depend on a new type of isoperimetric condition, which also admits some ($n{-}1$)-dimensional measures $\mu$, and exemplary configurations will be discussed. The long-term goal of these considerations is to extend the variational approach to prescribed mean curvature hypersurfaces in the spirit of Caccioppoli, De Giorgi, Miranda, Massari from $\mathrm{L}^1$ mean curvature to mean curvature given by a possibly lower-dimensional measure.

analysis of PDEs

Audience: researchers in the topic

Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php