Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs

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 -> infinity. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to infinity-Laplacians for scalar valued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al. (2014) for finding the zero of the infinity-Laplacian is given. Finally, we present applications in image inpainting.