Minimal Lipschitz Extensions for Vector-Valued Functions on Finite Graphs

摘要

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. We prove that the minimizers of functionals involving grouped e_p-norms converge to these extensions as p → ∞. Further, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to ∞-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed in [9] for finding the zero of the ∞-Laplacian is given.