@Semiclassical Any time I see the words "start with a parallel plate capacitor" I want to run away thanks to the horrific memories of undergrad finals :P
Maybe I can just write "clearly" and everyone will believe me
It's quite believable
but I reaaaaally don't want to do an $\epsilon-\delta$ proof if I can avoid it
It is easy to see that $\Gamma$ is $C^1$ in the Fr\'echet sense and that its partial derivative in the $u$ direction is given by $$D_1\Gamma(u,R)v=-a\Delta v-v-(2^\star-1)Ru^{2^\star-2}v,\quad v\in W^{2,p}.$$
@Semiclassical Of course, this one reminds me of the Astro fluid dynamics questions. It turned out that I wasn't able to do Astro fluid dynamics at all
@BalarkaSen Suppose I have a smooth function $f$ on a compact Riemannian manifold $M$. Can I find a smooth diffeomorphism $\phi$ of $M$ such that $f\circ\phi$ is close to $\inf f$ for a set of arbitrarily small comeasure?
Like, I want to smear out the region around $\inf f$ to be almost all of $M$
the goal being that $\|f\circ\phi-\inf f\|_{L^p}<\varepsilon$
@BalarkaSen In other words, can I find a diffeomorphism that takes a $\delta$-ball around a $p\in M$ to $U\subset M$ with measure$(M\setminus U)<\varepsilon$
@Slereah just take the red pill™ already... and see how far down the rabbit hole goes™! :P (ps wondering/ curious, how do you get that bohmian mechanics is a "mainstream" theory?)
