Visualization of the convex integration solutions to the Monge-Amp\`ere equation
Luca Codenotti, Marta Lewicka
TL;DR
This paper visualizes the initial iterations of convex integration solutions to the Monge-Ampère equation, illustrating the complex structures of approximate solutions in two dimensions.
Contribution
It implements an algorithm based on convex integration to visualize early-stage solutions to the Monge-Ampère equation, providing new insights into their structure.
Findings
Visualizations of the first iterations of convex integration solutions
Illustration of the Nash-Kuiper scheme in 2D
Approximation of anomalous solutions to the Monge-Ampère equation
Abstract
In this article, we implement the algorithm based on the convex integration result proved in [Lewicka-Pakzad Analysis and PDE (2017)] and obtain visualizations of the first iterations of the Nash-Kuiper scheme, approx- imating the anomalous solutions to the Monge-Amp\`ere equation in two dimensions.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Advanced Mathematical Physics Problems