Energy scaling law for a single disclination in a thin elastic sheet

We consider a single disclination in a thin elastic sheet of thickness $h$. We prove ansatz-free lower bounds for the free elastic energy in three different settings: First, for a geometrically fully non-linear plate model, second, for three-dimensional nonlinear elasticity, and third, for the F"oppl-von K'arm'an plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of $h$.