Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by:

$$ J(v) = \int_\Omega |v(x)|^p\mathrm{d}x $$ where $p <\frac{2n}{n-2}$. Let $v_0$ be the unique maximizer of $J$ over the set $\mathcal{A} = \{v \in \mathscr{H}_0^1 : \|v\|_{\mathscr{H}_1} \leq 1\}$ (the existence of this could be shown using, say, Rellich Kondrachov). Now, I would like to show which PDE this maximizer solves. The usual method is to compute the first variation and then from there obtain the Euler Lagrange Equation for the variational problem. However, this doesn't seem to work, as $J$ has no derivatives of $v$ in it. Specifically, if we fix some test function $\phi \in C_c^\infty(\Omega)$, and perturb by $t$ and compute the derivative, we get:

$$ \frac{d}{dt}\int_\Omega |(v + \phi t)|^p = \int_\Omega \frac{d}{dt}|v + (\phi t)|^p = \int_\Omega p|v + (\phi t)|^{p-1}\phi $$ (side note, why are we allowed to pass the derivative under the integral in the general case of the Lagrangian?). Evaluating the variation at time $t = 0$ and setting equal to $0$, we obtain: $$ \int_\Omega p|v|^{p-1}\phi = 0 $$ But this doesnt give us any useful information. What am I doing wrong here? How should I set up the variational problem?

  • $\begingroup$ Note the Euler-Lagrange formulation shouldn't work here. Maximisers at the boundary don't necessarily yield critical points and here it is obvious by scaling that the maximiser occurs at the boundary. This is demonstrated by the fact that your attempt only locates the unique minimiser $0$ as a critical point. Why do you expect the maximiser to solve some particular PDE? $\endgroup$ – Rhys Steele Mar 17 at 21:53
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    $\begingroup$ @RhysSteele right, it's fairly clear that $\|u\| = 1$ at the maximizer. The reason I expect this to occur is no reason of mine, but because it is what a (already submitted) homework question of mine stated. Perhaps I need to set this up as a constrained maximization? I.e through the use of Lagrange multipliers? $\endgroup$ – rubikscube09 Mar 17 at 21:57
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    $\begingroup$ Exactly; the right way of getting a PDE from that variational problem is Lagrange multipliers. Alternatively, consider the Euler-Lagrange equation for the ratio $J(v)/\|v\|_{H^1}$. You will end up with the same PDE. $\endgroup$ – Giuseppe Negro Mar 17 at 22:42
  • $\begingroup$ @GiuseppeNegro Thank you for the reply! I assume that Euler Lagrange is only useful then for computing minima/maxima on the interior of the admissible set $\mathcal{A}$, just like how the first order condition $\nabla f = 0$ works only for interior points in standard optimization over $\mathbb{R}^n$? $\endgroup$ – rubikscube09 Mar 17 at 22:51
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    $\begingroup$ Yes, that's it. Yours is a constrained optimization problem (unless you consider the ratio, as in my previous comment, , in which case it becomes a free optimization). For constrained optimization, use Lagrange multipliers. For free optimization, use Euler Lagrange. And yes, the same happens in finite dimension. $\endgroup$ – Giuseppe Negro Mar 17 at 23:23

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