Calculus of Variations and Geometric Measure Theory

P. Bella - R. Marziani

$\Gamma$-convergence for plane to wrinkles transition problem

created by marziani on 11 Dec 2023



Inserted: 11 dec 2023
Last Updated: 11 dec 2023

Pages: 47
Year: 2023


We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet, and identify the $\Gamma$-limit as the sheet thickness goes to $0$, thus extending the previous work of the first author Bella, ARMA 2015. The limiting problem is scalar and convex, but constrained and posed for measures. For the $\Gamma-\liminf$ inequality we first pass to quadratic variables so that the constraint becomes linear, and then obtain the lower bound using Reshetnyak's theorem. The construction of the recovery sequence for the $\Gamma-\limsup$ inequality relies on mollification of quadratic variables, and careful choice of multiple construction parameters. Eventually for the limiting problem we show existence of a minimizer and equipartition of the energy for each frequency.