Analytical properties for degenerate equations

TL;DR

This survey explores how solutions to certain degenerate elliptic equations, despite low regularity, exhibit properties similar to analytic functions, shedding light on their underlying structure and related open problems.

Contribution

It demonstrates that solutions to a key degenerate elliptic equation possess analytic-like properties despite lacking high regularity.

Findings

Solutions exhibit properties akin to analytic functions.

Degenerate elliptic equations have solutions with unexpected regularity features.

Insights may inform open problems in PDE regularity theory.

Abstract

By a classical result, solutions of analytic elliptic PDEs, like the Laplace equation, are analytic. In many instances, the properties that come from being analytic are more important than analyticity itself. Many important equations are degenerate elliptic and solutions have much lower regularity. Still, one may hope that solutions share properties of analytic functions. These properties are closely connected to important open problems. In this survey, we will explain why solutions of an important degenerate elliptic equation have analytic properties even though the solutions are not even $C^3$.