Bernstein theorems for nonlinear geometric PDEs
Connor Mooney
TL;DR
This paper reviews Bernstein theorems for various nonlinear geometric PDEs, discussing classical and recent results across minimal surfaces, Monge-Ampère, and special Lagrangian equations, based on a lecture series.
Contribution
It provides an expository overview of Bernstein theorems for multiple nonlinear geometric PDEs, highlighting recent developments and open problems.
Findings
Bernstein theorems hold for certain classes of nonlinear geometric PDEs.
The paper discusses extensions and limitations of classical results.
Connections between different geometric PDEs are explored.
Abstract
In this expository article we revisit the Bernstein problem for several geometric PDEs including the minimal surface, Monge-Amp\`{e}re, and special Lagrangian equations. We also discuss the minimal surface system where appropriate. The article is based on a lecture series given by the author for the inaugural European Doctorate School of Differential Geometry, held in Granada in June 2024.
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Taxonomy
TopicsNonlinear Differential Equations Analysis · Numerical methods for differential equations · Mathematical and Theoretical Analysis