On the definitions of Sobolev and BV spaces into singular spaces and the trace problem

The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to N. Korevaar and R. Schoen on the one hand, and J. Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to L. Ambrosio (for BV maps into metric spaces), Y.G. Reshetnyak and finally to the notion of Newtonian-Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of J. Bourgain, H. Brezis and P. Mironescu in terms of “limit” of the space W s,p as s → 1, 0 < s < 1, and finally following the approach proposed by H.M. Nguyen. We also establish the W s− 1 p ,p regularity of traces of maps in W s,p (0 < s ≤ 1 < sp).