Korevaar–Schoen’s energy on strongly rectifiable spaces

Abstract We extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case source space being an $$\mathsf{RCD}$$ RCD space. In this situation it appears that no version ‘subpartition lemma’ holds: obtain both existence limit approximated energies and lower semicontinuity energy we shall rely on: fact such spaces are ‘strongly rectifiable’ a notion which is first-order in nature (as opposed measure-contraction-like properties, second order). This particularly useful combination with Kirchheim’s differentiability theorem, as allows approximate result turn quickly provides representation for density, differential calculus developed by first author allows, thanks formula prove here, desired from closure abstract differential. When target $$\mathsf{CAT}(0)$$ CAT ( 0 ) can also identify density Hilbert-Schmidt norm differential, line smooth situation.