P-Harmonic mappings between metric spaces

Abstract

In this paper, we solve the Dirichlet problem for Sobolev maps between singular metric spaces that extends the corresponding result of Guo and Wenger (Commun Anal Geom 28(1):89–112, 2020). The main new ingredient in our proofs is a suitable extension of the theory of trace for metric valued Sobolev maps developed by Korevaar and Schoen (Commun Anal Geom 1(3–4):561–659, 1993) . We also develop a theory of trace in the borderline case, which investigates a sharp condition to characterize the existence of traces.