Curvature energies for non-smooth subsets of Euclidean spaces

• Funding agency: Polish National Science Center (NCN)
• Grant type: HARMONIA
• Grant number: 2013/10/M/ST1/00416
• Years: 2014 -- 2017
• Principal investigator: prof. Paweł Strzelecki
• Total funding amount: 425 880 PLN

This project is devoted to several topics in geometric analysis and its applications to other branches of mathematics and to the natural sciences. Its main purpose is to study non-smooth (countably rectifiable or just measurable) m-dimensional subsets $\mathrm{\Sigma }\subseteq {\mathbf{R}}^n$ by means of so-called geometric curvature energies, i.e., integral functionals defined geometrically but without referring a priori to any smoothness of $\mathrm{\Sigma }$. One possibility is to use the Lp–norms (with respect to the m-dimensional Hausdorff measure on $\mathrm{\Sigma }$) of functions that are defined in metric terms and designed so as to penalize self-intersections and vicinity of intrinsically distant points. In the simplest case of rectifiable curves in ${\mathbf{R}}^3$ one such energy is the integral Menger curvature ${\mathcal{M}}_p$ of a rectifiable curve $\gamma$: the triple integral of $1/R^p$, where $R=R(x,y,z)$ is the radius of the circumcircle of the points $x$, $y$ and $z$ lying on the curve, and each integration is performed with respect to the arc length. Recently, because of the striking connections with many branches of pure mathematics (harmonic analysis, variational calculus, geometric knot theory) and with applications for modelling of the physical objects without self-intersections (continuum elastic rods avoiding self-contact, membranes, entangled or knotted DNA molecules, optimal packing issues etc.), similar research is being conducted in many centers worldwide.