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Abstract:   We propose to study some mathematical problems, combining analysis and pdes, that arise from questions of thin film materials design and the so-called prestrained elasticity. We will see how the analysis of scaling of the energy minimizers in terms of the film's thickness leads to the rigorous derivation of a hierarchy of limiting theories, differentiated by the embeddability properties of the target (prestrain) metrics and, a-posteriori, by the emergence of isometry constraints with low regularity. This leads to questions of rigidity and flexibility of solutions to the weak formulations of the related pdes, including the Monge-Ampere equation. We will show how the Nash-Kuiper convex integration can be applied here to achieve flexibility of Holder solutions, and how other techniques from fluid dynamics (the commutator estimate, yielding the degree formula in the  present context) are 
useful  in proving the rigidity of Holder solutions. We also implement the algorithm based on the convex integration result and obtain visualizations of the first iterations approximating the anomalous solutions to the Monge-Ampere equation in two dimensions.