Notes on the Infinity-Laplace Equation

TL;DR

This paper discusses the properties, mathematical background, and applications of the infinity-Laplace equation, highlighting its connections to classical PDEs and its relevance in image processing and mass transfer problems.

Contribution

It provides an overview of the infinity-Laplace equation, emphasizing its mathematical properties, classical analogs, and practical applications, serving as a foundational reference.

Findings

Connections to Dirichlet integral and Mean Value Theorem

Applications in image processing and mass transfer

Provides optimal Lipschitz extensions

Abstract

These notes are written up after my lectures at the University of Pittsburgh in March 2014 and at Tsinghua University in May 2014. My objective is the $\infty$-Laplace Equation, a marvellous kin to the ordinary Laplace Equation. The $\infty$-Laplace Equation has delightful counterparts to the Dirichlet integral, the Mean Value Theorem, the Brownian Motion, Harnack's Inequality and so on. It has applications to image processing and to mass transfer problems and provides optimal Lipschitz extensions of boundary values. My treaty of this "fully non-linear" degenerate equation is far from complete and generalizations are deliberately avoided.