Is it possible to determine the best constant $S(\Omega, p)$ of the embedding $H^1_{0, r}(\Omega)$ to $L^{p+1}(\Omega)$ where $p>1$ and $\Omega=\{a<x<b\}.$
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$\begingroup$ duplicate (unanswered): math.stackexchange.com/questions/2612409/… $\endgroup$ – macbeth Jan 23 at 14:35

$\begingroup$ A reasonable initial approach is to assume that the extremal function is a function of $x$. In that case, the function has to satisfy a second order ODE and of course, vanish at $x =a$ and $x=b$. Now the question is whether there is a closed form solution to this or the constant can be computed indirectly. You might be able to narrow down what the function is by looking at what happens under rescaling of the domain as well as the inversion map $y = x/x^2$. A reasonable thing to do is to assume $a = 1$, so the extremal function depends only on the parameter $b$. $\endgroup$ – Deane Yang Jan 23 at 18:51

1$\begingroup$ @sgdas please add the definition of the space $H^1_{0,r}(\Omega)$ and its norm. $\endgroup$ – Pietro Majer Feb 2 at 20:16