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Actually this is one of the oldest problems in the calculus of variations. It's named "the Newton problem" after Sir Isaac Newton, who studied it in 1685. It arises from the determination of the optim …

Certainly, a piecewise affine function $f$ (meaning, a function which is affine on each open simplex of some triangulation of the domain) is in $W^{1,\\ p}_{loc}$ for some $1\leq p\leq \infty$ if and …

Especially in Calculus of Variations and Mechanics, a submanifold $Y$ of a manifold $X$ is usually called "a natural constraint" for a functional $f$ on $X$, if the special circumstance that you are c …

The length of these curves is unbounded. For a positive integer $n$ consider a triangular wave $f_n:[0,1]\to\mathbb{R}$ with support on $[1/3,2/3]$, making $n$ (isosceles) triangular impulses on $[1 …

I'd say these belongs to the family of shape optimization problems in the Calculus of Variations (of whom the most famous example is the isoperimetric problem). As to the first one, you should make s …

This is elementary but let's state it. First one only considers smooth test functions $h$ with compact support (so that the last two terms disappear). If $F(x):=\int_0^xf(t)dt$, integrating by parts t …

In fact it's just the MVT to $DF$: $$\|DF(x_n)-DF(y_n)\|\le \|x_n-y_n\|\sup\|D^2F\|=O(\|x_n-y_n\|)=o(1),$$ so $\|DF(x_n)\|=o(1)$ too. $$*$$ Note that if $D^2f$ is not bounded, it is not true, and (a …

It seems to me that the set of projective semi-distances that you describe in the example (let's call them of type I) can be slightly generalized the following way. In the definition of $d_H$, let $H …

As your title suggests, the dominated convergence theorem is a powerful tool, which often we can apply to cases to where the assumption of domination is not directly verified, or nor immediate to ver …

I think everything is clear to you by now, anyway here are some elementary fact that seem relevant. For an open set $\Omega\subset \mathbb R^n$, $f\in W^{1,\infty}(\Omega)$ iff the restrictions $f_{| …