In this talk, we will present results regarding the regularity and
rigidity of solutions to the Monge-Ampère equation, inspired by
the role played by this equation in the context of prestrained
elasticity. 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) find their parallels in proving
the rigidity of Holder, Sobolev or Besov solutions. A prestrained
elastic body is a three dimensional elastic object, modeled in its
reference configuration by an open domain and a Riemannian metric.
This metric, by the main ansatz, is induced by mechanisms such as
growth, plasticity, thermal expansion etc, and is the cause of the
elastic deformation that determines the shape of the body. One can
seek this deformation through a variational minimization principle, as
the best possible immersion of the given Riemannian manifold into the
flat space. For thin films, the characteristics of the original
prestrain metric induce the distinct nonlinear theories in the
singular limit of vanishing thickness, that determine the above
mentioned minimizing deformation. A particular aspect of these limit
theories are the curvature constraints that are manifested as the
Monge-Ampère equation in the appropriate energy regimes.
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