Does there exist energy-minimizing immersions? This is a cross-post.
Let $M,N$ be $d$-dimensional oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E_d:C^{\infty}(M,N) \to \mathbb{R}$ be the $d$-energy, i.e.
$$ E_d(f)=\int_M |df|^d \text{Vol}_M.$$
Set $$E_{M,N}=\inf \{ E_d(f) \, | \,\, f \in C^{\infty}(M,N) \text{ is an immersion} \}.$$
Is $E_{M,N}$ always a minimium? i.e. does there exist an immersion with minimal energy?
(I am assuming there exist at least one immersion from $M$ to $N$.  )
I am specifically considering the $d$-energy between $d$-manifolds, and not 
 the more classical $2$-energy. This is because in the case of the $2$-energy the answer can be negative in general; it is well-known that
$$\inf_{f \in \text{Diff}(\mathbb{S}^n) }  E_2(f) =0$$ when $n >2$,
but of course there is no immersion with zero $2$-energy.
However, the identity map $\text{Id}_{M^d}$ has minimal $d$-energy among all diffeomorphisms.
 A: 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 $A_*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If 
$$
\frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right)
$$
then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is 
$$
h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\
\frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R,
\end{cases}
$$
where $\sigma$ is defined by
$$
\frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right).
$$
See Theorem 1.8 in  T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).
It seems you get the same limit if you restrict the minimization problem to diffeomorphisms with given boundary data.
Tadeusz Iwaniec (not Henryk Iwaniec) wrote many papers regarding minimization of the n-energy so you should check his recent publications to see if you find there relevant results.
