Analysis Seminar

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Elasticity with residual stress and related problems

Speaker: Marta Lewicka, University of Pittsburgh

Location: Warren Weaver Hall 1302

Date: Thursday, February 22, 2018, 11 a.m.

Synopsis:

We study some mathematical problems in Analysis and Pdes, that arise from
questions of thin film shape formation and the so-called prestrained
elasticity. We will see how the scaling of the energy minimizers in terms of
the film's thickness leads to the 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.