Let $M,N$ be $d$-dimensional oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E:C^{\infty}(M,N) \to \mathbb{R}$ be the Dirichlet energy, i.e.
$$ E(f)=\int_M |df|^2 \text{Vol}_M.$$
For any given homotopy class $\alpha$ of smooth maps $M \to N$, define $$E_{\alpha}=\inf \{ E(f) \, | \,\, f \in \alpha \}.$$
(In some cases, we know $E_{\alpha}$ is realized, e.g. when $N$ is closed and of negative curvature.)
Now define $$E:=\inf_{\alpha \text{ is not trivial }} E_{\alpha}$$
Is $E$ always a minimium? i.e. does there exist a homotopy class $\alpha$ such that $E=E_{\alpha}$?
Does the answer change if we replace the Dirichlet energy with the $d$-energy? (i.e. integrate $|df|^d$ instead of $|df|^2$?)
In the case $M=N=\mathbb{S}^1$, one can prove directly, via Holder's inequality, that the answer is positive:
$$ \big(\deg f \cdot \text{Vol}(\mathbb{S}^1)\big)^2=\big(\int_{\mathbb{S}^1} |f'|\big)^2 \le \int_{\mathbb{S}^1} |f'|^2 \int_{\mathbb{S}^1} 1^2=\text{Vol}(\mathbb{S}^1)E(f),$$ so we get
$$ E(f) \ge (\deg f)^2 \cdot \text{Vol}(\mathbb{S}^1),$$
and for a map of degree $n$, the minimal energy is obtained exactly for $f(e^{i\theta})=e^{in\theta}$.
So, the non-trivial homotopy class with minimal infimal energy is the class of degree-one maps.
More generally, if we consider the $d$-energy, and assume $M,N$ are closed, then
$$ E_d(f)=\int_M |df|^d \text{Vol}_M \ge d^{\frac{d}{2}} \int_M \det df \text{Vol}_M=d^{\frac{d}{2}} \int_{M} f^*\text{Vol}_N=d^{\frac{d}{2}} \deg f \int_{N} \text{Vol}_N,$$ so $$ E_d(f) \ge d^{\frac{d}{2}} \deg f \cdot \text{Vol}(N),$$ with equality if and only if $f$ is conformal. I am not sure this helps to settle the question though.