Lipschitz Extension Problem

Abstract

This project is mainly concerned with the understanding of the problem of extending Lipschitz functions from the boundary of some set to inside it in some special way. This extension is minimal/least fluctuating in some sense and is called the Absolutely Minimizing extension(AM). It has applications in fields like image processing and analyzing the shape of sandpiles where one can interpolate with incomplete information and fill the gaps. This is interesting from a theoretical standpoint because some of the core topics in PDE like viscosity solutions and variational methods are useful to arrive at some of the important results concerning existence and uniqueness. Interestingly, AM are viscosity solutions of the infinity Laplacian ($\Delta_{\infty}u=0)$ which is a quasilinear second-order degenerate pde.