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I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A_*=A(r …

Existence of minimizers should not be a severe issue... The proof of the existence follows form the Arzela-Ascoli theorem, but the proof is not entirely obvious. This is Proposition 1.1 in: E. G …

This is true and it follows from the following result: Lemma. Suppset that $f:W\to\mathbb{R}$ is convex and $L$-Lipschitz, where $W\subset\mathbb{R}^n$ is convex. Then, $$ \tilde{f}(x)=\inf_{z\in W}\ …

You actually do not need to assume that the mappings are Lipschitz as it is true for general $W^{1,n}$ mappings Theorem. If $\mathcal{M}$ and $\mathcal{N}$ are smooth compact and oriented manifolds, …

Example 7.7 in L. Ambrosio, A. Coscia, Alessandra, G. Dal Maso, Fine properties of functions with bounded deformation. Arch. Rational Mech. Anal. 139 (1997), no. 3, 201–238.

There is a much more general result of Vallée-Poussin from which a negative answer to your question follows. Let $(X,\mu)$ be a measure space. We say that a family of function $\mathcal{F}\subset L^1( …