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Without controlling the derivative, the problem is a bit ill posed, as $x(T)$ can take any value independent of what $x$ does in the rest of the integral. So your solution will be a minimizer of $\int …

In general, a first variation is just the collection of all directional derivatives $\frac{d}{d\epsilon} \mathcal{F}(\rho+\epsilon\chi)|_{\epsilon = 0}$. For fixed $\rho$ one can treat them as a funct …

The following is a classical definition due to Morrey: Let $\Omega \subset \mathbb{R}^n$ be a nice enough, bounded domain and $f: \mathbb{R}^{m \times n} \to \mathbb{R}$ with some reasonable growth co …

There is the following interpretation coming from physics and continuum mechanics, which is a bit too long for a comment but might be helpful: If you think of $\mathcal{F}$ as an energy that you want …

You can treat this as a problem with two Lagrange multipliers. Then by standard methods, a minimizer $f$ has to exist (by convexity in $f'$) and has to be a weak solution to $$-f'' + \lambda f + \mu f …

This is not a full answer, since I do not know the counterexample Lin refers to, but I can offer some explanations and guesses which are too long for a comment: You can define a first variation for cu …

There is a counterexample, however there might be ways to avoid it. Take $\mathcal{M} = \mathcal{N} =\mathbb{S}^2$, but now consider sequence of maps that cover the sphere twice, where you shrink the …

Okay, let me try a writeup of the comment chain. For any reasonable subset $A\subset \Omega_2$ and $B := f^{-1}(A)$ you get $$\int_A |f^{-1}(y)| dy = \int_B \det df dx \leq \liminf_{n\to\infty} \int_B …