# Analysis Seminar

#### 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.