On the Equation (∇u)^tH(u)(∇u) = G

Abstract: 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 immersionu: Ω→R^n of appropriate regularity, satisfying (∇u)^tH(u)(∇u) =G in Ω......(1)

We consider Cauchy, Dirichlet and Dirichlet-Neumann problems.Equation (1) comes up in diverse contexts. WhenH(and henceG) issymmetric and positive definite, Equation (1) is connected to the prob-lem of equivalence of Riemannian metrics. The symmetric case is alsoimportant in the non-linear elasticity theory because of its connectionwith the Cauchy-Green deformation tensor. WhenH(and henceG) isskew-symmetric, Equation (1) comes up in the context of the problemof equivalence of differential two-forms.The aim of the talk is to present a survey of recent progress and ad-vances made in the context of Equation (1). We shall also discuss thegeneral case whenH,Gare neither symmetric nor skew-symmetric.

analysis of PDEsdynamical systemsfunctional analysisoptimization and controlspectral theory

Audience: general audience

Webinar on PDE and related areas

Series comments: The webinar is organised jointly from IIT-Kanpur, TIFR-CAM,Bangalore, IISER-Pune and IISER-Kolkata.

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