Korevaar–Schoen’s energy on strongly rectifiable spaces

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

Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows

We develop Korevaar–Schoen’s theory of directional energies for metric-valued Sobolev maps in the case of $${\textsf {RCD}}$$ RCD source spaces; to do so we crucially rely on Ambrosio’s concept of Regular Lagrangian Flow. Our review of Korevaar–Schoen’s spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of ‘differential of a map along a vector field’ and about the parallelogram identity for $${\textsf {CAT}}(0)$$ CAT ( 0 ) targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.

Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps

A free homotopy decomposition of any continuous map from a compact Riemannian manifold 𝓜 to a compact Riemannian manifold 𝓝 into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in Ws,p(𝓜, 𝓝), with sp = m = dim 𝓜. In particular, when the fundamental group π1(𝓝) acts trivially on the homotopy group πm(𝓝), the number of homotopy classes to which a map can belong can be estimated by its Sobolev energy. The estimates are particular cases of estimates under a boundedness assumption on gap potentials of the form $$\begin{array}{} \displaystyle \iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}} \frac{1}{d_\mathcal{M} (y, x)^{2 m}} \, \mathrm{d} y \, \mathrm{d}x. \end{array}$$ When m ≥ 2, the estimates scale optimally as ε → 0. When m = 1, the total variation of the maps appearing in the decomposition can be controlled by the gap potential. Linear estimates on the Hurewicz homomorphism and the induced cohomology homomorphism are also obtained.

Global invertibility of Sobolev maps

We define a class of Sobolev {W^{1,p}(\Omega,\mathbb{R}^{n})} functions, with {p>n-1} , such that its trace on {\partial\Omega} is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticity.