PDE seminar: Poincaré Path Integrals for Elasticity
Speaker: Espen Sande (UiO)
Title: Poincaré Path Integrals for Elasticity
Abstract: In three space dimensions it is well known that if a smooth vector field u satisfies curl u = 0 it is, on a contractible domain, the gradient of a scalar function φ, i.e., grad φ = u. This function φ can be recovered from an explicit path integral of u. More generally, this fact follows from the existence of null-homotopies, or Poincaré operators, for the de Rham complex.
In this talk we consider the elasticity complex and prove that there exist Poincaré operators for it that satisfy many important properties. We also provide the explicit path integrals for these operators. Our approach is to use the Poincaré path integrals for the de Rham complex together with the Bernstein-Gelfand-Gelfand (BGG) resolution. This generalizes a classical result in the theory of linear elasticity that dates back to the work of Cesàro in 1906 and Volterra in 1907.
This is joint work with Snorre H. Christiansen and Kaibo Hu.