The H^-1-norm of tubular neighbourhoods of curves

2009/03/22 by Yves van Gennip, van Gennip, Yves, Mark A. Peletier +1
Computer Science · Mathematics · Engineering · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Composite Material Mechanics

paper · pdf · doi:10.48550/arxiv.0903.3709

Abstract

We study the H^-1-norm of the function 1 on tubular neighbourhoods of curves in R². We take the limit of small thickness epsilon, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit epsilon to 0, containing contributions from the length of the curve (at order epsilon³), the ends (epsilon⁴), and the curvature (epsilon⁵). The second result is a Gamma-convergence result, in which the central curve may vary along the sequence epsilon to 0. We prove that a rescaled version of the H^-1-norm, which focuses on the epsilon⁵ curvature term, Gamma-converges to the L²-norm of curvature. In addition, sequences along which the rescaled norm is bounded are compact in the W^1,2 -topology. Our main tools are the maximum principle for elliptic equations and the use of appropriate trial functions in the variational characterisation of the H^-1-norm. For the Gamma-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.

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