TY - JOUR
T1 - The H^{-1}-norm of tubular neighbourhoods of curves
AU - Gennip, van, Y.
AU - Peletier, M.A.
PY - 2011
Y1 - 2011
N2 - We study the H^{-1}-norm of the function 1 on tubular neighbourhoods of curves in R^2. 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 epsilon^3), the ends (epsilon^4), and the curvature (epsilon^5). 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 epsilon^5 curvature term, Gamma-converges to the L^2-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.
AB - We study the H^{-1}-norm of the function 1 on tubular neighbourhoods of curves in R^2. 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 epsilon^3), the ends (epsilon^4), and the curvature (epsilon^5). 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 epsilon^5 curvature term, Gamma-converges to the L^2-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.
U2 - 10.1051/cocv/2009044
DO - 10.1051/cocv/2009044
M3 - Article
SN - 1292-8119
VL - 17
SP - 131
EP - 154
JO - ESAIM : Control, Optimisation and Calculus of Variations
JF - ESAIM : Control, Optimisation and Calculus of Variations
ER -