Isometric embeddings and scaling laws for compressed elastic sheets

The elastic energy of a thin elastic sheet is the sum of a non-convex term that penalizes local stretching (and compression) and of a small (depending on the thickness) singular perturbation that takes bending into account.

When the sheet is confined in a region of small diameter, these two terms enter in competition and a concentration of curvature on lines and points is observed.

By using a combination of explicit constructions and general results from differential geometry we obtain an upper bound for the elastic energy of a compressed sheet in terms of its thickness at the power $5/3$. We show that this exponent is optimal in certain simplified geometries, that are conjectured to represent the canonical singularities leading to this exotic power law.