Seminari del Dipartimento di Matematica

Seminario del 2023

Saugata Bandyopadhyay

On the Equation (∇u)^{t}H(u)∇u = G

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

Let n ∈ N, n > 2 and let Ω ⊆ R^n be open. Let H, G : R^n → R^{n×n} be of appropriate regularity. We discuss the existence of an immersion u : Ω → R^n of appropriate regularity, satisfying (∇u)^tH(u)(∇u) = G in Ω. (1) We consider both local and global problems. Equation (1) comes up in diverse contexts. When H (and hence G) is symmetric and positive definite, Equation (1) is connected to the problem of equivalence of Riemannian metrics. The symmetric case is also important in the non-linear elasticity theory because of its connection with the Cauchy-Green deformation tensor. When H (and hence G) is skew-symmetric, Equation (1) comes up in the context of the problem of equivalence of differential two-forms. The aim of the talk is to present a survey of recent progress and advances in the context of Equation (1). We also discuss the general case when H, G are neither symmetric nor skew-symmetric. The talk is based on joint works with Bernard Dacorogna, Vladimir Matveev and Marc Troyanov.