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Energy scaling law for the regular cone

We consider a thin elastic sheet in the shape of a disk whose reference metric is that of a singular cone. I.e., the reference metric is flat away from the center and has a defect there. We define a geometrically fully nonlinear free elastic energy, and investigate the scaling behavior of this energy as the thickness $h$ tends to 0. We work with two simplifying assumptions: Firstly, we think of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space and assume that the exponential map at the origin (the center of the sheet) supplies a coordinate chart for the whole manifold. Secondly, the energy functional penalizes the difference between the induced metric and the reference metric in $L^\infty$ (instead of, as is usual, in $L^2$). Under these assumptions, we show that the elastic energy per unit thickness of the regular cone in the leading order of $h$ is given by $C^*h^2|\log h|$, where the value of $C^*$ is given explicitly.