Analysis Seminar: Itai Alpern (HUJI) — Asymptotic rigidity for shells in non-Euclidean elasticity

Wed, 12/05/202112:00-13:00
Title: Asymptotic rigidity for shells in non-Euclidean elasticity

Abstract: An elastic shell can be modeled as a 2-dimensional Riemannian manifold (M,g)​​ endowed with a "reference second fundamental form" b​​
The elastic energy E(f)​​ of an embedding f: (M,g) → ℝ3​​ then measures the deviation of the first and second fundamental forms of the embedded surface from the reference ones.
A classical result asserts that if g and b satisfy some compatibility conditions — the Gauss–Codazzi equations — then there exists a configuration f with zero elastic energy.
However, it is not clear that E​​ attains its minimum. It is therefore natural to ask whether inf E = 0​​ implies that g​​ and b​​ are compatible, and in this case, whether any minimizing sequence converges to a zero-energy configuration.
In this lecture I will show that this is indeed true, as well as generalizations of this result. I will present a sketch of the proof of this theorem, emphasizing its main ideas.

Based on a joint work with Raz Kupferman and Cy Maor.