Viscosity Solution of the Infinity Laplacian Equation

Abstract

Nonlinear elliptic partial differential equations arise in many areas of analysis and applied mathematics. Among them, the p-Laplace equation which appears as the Euler–Lagrange equation associated with the minimization of the p-Dirichlet functional. For finite p, solutions of the p-Laplace equation admit a weak interpretation. As p tends to infinity these variational problems converge to a limiting equation known as the infinity-Laplacian. Unlike the p-Laplacian, the infinity-Laplacian must be interpreted in the viscosity sense rather than the weak sense. The main objective of this thesis is to analyze the convergence of weak solutions of the p-Laplace equation to viscosity solutions of the infinity-Laplace equation as p tends to infinity.

The approach relies on tools used from Sobolev space theory, variational methods, compactness arguments, functional analysis and non-linear PDE analysis.