On the analyticity of solutions to non-linear elliptic partial differential equations

TL;DR

This paper provides a straightforward proof that smooth solutions to second-order non-linear elliptic PDEs are analytic, utilizing inductive estimates and Cauchy's method of majorants.

Contribution

It offers an accessible proof of the analyticity of solutions to non-linear elliptic equations, building on Kato's ideas and simplifying previous approaches.

Findings

Solutions to non-linear elliptic PDEs are analytic if they are smooth.

The proof employs inductive estimates for weighted derivatives.

Cauchy's method of majorants is used to conclude analyticity.

Abstract

We give an easy proof of the fact that $C^\infty$ solutions to non-linear elliptic equations of second order $$ \phi(x, u, D u, D^2 u)=0 $$ are analytic. Following ideas of Kato, the proof uses an inductive estimate for suitable weighted derivatives. We then conclude the proof using Cauchy's method of majorants}.