Hebrew University topology and geometry seminar

November 11, 2015
Ross building, Seminar Room 70A

Cy Maor

Hebrew University of Jerusalem

Limits of elastic energies of converging Riemannian manifolds

Abstract: An elastic energy functional of a Riemannian manifold $\mathcal{M}$ is a function that measures the distance of an embedding $u: \mathcal{M} \to \mathbb{R}^d$ from being isometric. In many applications, the manifold in consideration is actually a limit of other manifolds, that is, $\mathcal{M}$ is a limit of $\mathcal{M}_n$ in some sense. Assuming that we have an elastic energy functional for each $\mathcal{M}_n$, can we obtain an energy functional of $\mathcal{M}$ which is a limit of the functionals of $\mathcal{M}_n$?
In this talk I will show that for prototypical elastic energies and an appropriate notion of convergence of manifolds we can indeed obtain a limit elastic energy (in the sense of $\Gamma$-limits of functionals), and will give an upper bound for a limit for a weaker notion of convergence. If time permits, I will discuss the relevance of the results to the theory of dislocations, and present some open questions.
The talk is based on an ongoing work with Raz Kupferman. No knowledge in elasticity theory or in $\Gamma$-convergence will be assumed.