I came across this blog that says that its French version has answers to most of Arnol'd's trivium problems, and I figured I'd try my hand at some of the ones they don't have. Number 68 raised my interest, and I think I solved it, but I'm not very happy with my solution and I'd like to know whether there's a more elegant approach, more based on general principles and less dependent on hacking together a somewhat arbitrary function. I've written up my solution as an answer, but I'm hoping for other, perhaps better answers. Here's the problem:
Find $$ \inf \iint\limits_{x^2+y^2\leqslant1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\mathrm dx\,\mathrm dy $$
for $C^\infty$-functions $u$ that vanish at $0$ and are equal to $1$ on $x^2+y^2=1$.