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On the viscosity linearization method without compactness

Anthony Salib

Abstract

Savin's small perturbation approach has had far reaching applications in the theory of non-linear elliptic and parabolic PDE. In this short note, we revisit his seminal proof of De-Giorgi's improvement of flatness theorem for minimal surfaces and provide an approach based on the Harnack inequality that avoids the use of compactness arguments.

On the viscosity linearization method without compactness

Abstract

Savin's small perturbation approach has had far reaching applications in the theory of non-linear elliptic and parabolic PDE. In this short note, we revisit his seminal proof of De-Giorgi's improvement of flatness theorem for minimal surfaces and provide an approach based on the Harnack inequality that avoids the use of compactness arguments.
Paper Structure (7 sections, 9 theorems, 72 equations)

This paper contains 7 sections, 9 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Structure of the proof
  3. Acknowledgements
  4. Notation and Preliminaries
  5. Notation
  6. Preliminaries
  7. Proof of Theorem \ref{['improvementofflatness']}

Key Result

Theorem 1.1

There exists universal constants $\varepsilon_1(n)$ and $\eta > 0$ such that if $\partial E$ is a minimal surface and for some $\varepsilon \leq \varepsilon_1(n)$, then

Theorems & Definitions (19)

  • Theorem 1.1: Harnack Inequality
  • Theorem 1.2: Improvement of Flatness
  • Definition 2.1
  • Definition 2.2: Viscosity solution
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 9 more