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Calculus of Variations and Geometric Measure Theory

P. Bella - R. Marziani

Γ-convergence for plane to wrinkles transition problem

created by marziani on 11 Dec 2023
modified on 12 Nov 2024

[BibTeX]

Accepted Paper

Inserted: 11 dec 2023
Last Updated: 12 nov 2024

Journal: Journal of Nonlinear Science
Pages: 51
Year: 2023

Abstract:

We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet, and identify the Γ-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 Γlim inf 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.


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