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

Simon Blatt

Abstract

We give an easy proof of the fact that $C^\infty$ solutions to non-linear elliptic equations of second order $$ φ(x, u, D u, D^2 u)=0 $$ 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}.

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

Abstract

We give an easy proof of the fact that solutions to non-linear elliptic equations of second order 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}.

Paper Structure

This paper contains 11 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterisation of analytic functions
  4. $L^2$-estimates
  5. A Banach algebra
  6. Binomials
  7. Higher order chain rule
  8. The proof
  9. Reduction of the problem
  10. A recursive inequality
  11. Cauchy's method of majorants

Key Result

Theorem 1.1

Let $u\in C^\infty (\Omega, \mathbb R)$ be a solution to eq:PDE, $x_0 \in \Omega$, and $\phi$ be analytic in a neighborhood of $(x_0,u(x_0), Du(x_0), D^2(x_0)$. Then $u$ is real analytic near $x_0$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5: Adams1975
  • Remark 2.6
  • Proposition 2.7: Kato1996
  • ...and 8 more