# Minimal Lipschitz and $\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs

## TL;DR

This paper investigates minimal Lipschitz and $
abla$-harmonic extensions of vector-valued functions on finite graphs, showing convergence of $p$-Laplacian solutions to these extensions and exploring their applications in image inpainting.

## Contribution

It establishes the equivalence of lex and L-lex minimal extensions, proves convergence of $p$-Laplacian solutions to minimal Lipschitz extensions, and connects these concepts to iterative filters and $
abla$-Laplacians.

## Key findings

- Lex and L-lex minimal extensions are identical.
- Solutions of graph $p$-Laplacians converge to minimal Lipschitz extensions as $p o
abla$.
- Iterative algorithms for $
abla$-Laplacians are proven to converge.

## Abstract

This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz extensions. Then we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\to \infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to $\infty$-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al.~(2014) for finding the zero of the $\infty$-Laplacian is given. Finally, we present applications in image inpainting.