Scaling laws and limit models for non-Euclidean Elasticity functionals

It is well known that a smooth Riemannian metric on a simply
connected domain can be realized as the pull-back metric of an orientation
preserving deformation if and only if the associated Riemann curvature
tensor vanishes identically. When this condition fails, one could seek a
deformation yielding the closest metric realization. In this talk, we set
up a variational formulation of this problem by introducing a
non-Euclidean version of nonlinear elasticity functionals. The
corresponding variational problem describes elastic structures and growing
tissues (leaves, flowers or marine invertebrates) which exhibit non-zero
strain at free equilibria.

We discuss the scaling laws and $\Gamma$-limits of the introduced 3d
functional on thin plates in the limit of vanishing thickness. One
important feature is the study of Sobolev spaces of isometries and
infinitesimal isometries. In particular, we obtain an equivalent condition
for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric
into $\mathbb R^3$.