This lecture will be concerned with the analysis of thin elastic films which exhibit residual stress at free equilibria. Examples of such structures include, in particular, growing tissues (such as leaves) as well as specifically engineered gels.

There, the growth process results in the formation of non-Euclidean target metrics, leading to complicated morphogenesis of the tissue which attains a configuration closest possible to an isometric immersion of the metric. This phenomenon can be studied through a variational model, pertaining to the non-Euclidean version of the nonlinear elasticity.

We will explain the basic model and the analytical problems which emerge from its study.