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does anyone know a reference where I can find the best constant for Sobolev inequality on an annulus?

More precisely, I know by Rellich-Kondrachov Theorem (Brezis, Theorem 9.16) that, given $\Omega\subset\mathbb{R}^2$ bounded and of class $C^1$, we have the embedding \begin{equation} W^{1,2}(\Omega)\subset L^q(\Omega),\quad\forall q\in[p,+\infty). \end{equation} This is equivalent to \begin{equation} \|u\|_{L^q(\Omega)}\le C(q,\Omega) \|u\|_{W^{1,2}(\Omega)}. \end{equation} I was wondering what is known about $C(q,\Omega)$ in case $\Omega$ is the annulus \begin{equation} A_x(\delta,\tau):=\{y\in\mathbb{R}^2,\,|\,\delta\le|x-y|\le\tau\}. \end{equation} Thank you very much!

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    $\begingroup$ you might just retsart the proof given in Brezis by keeping track of your constant $\endgroup$ – Guy Fsone Jan 19 '18 at 17:22
  • $\begingroup$ Yes, it is certainly a possibility. However, I fear I would hardly get the sharp constant $\endgroup$ – popoolmica Jan 20 '18 at 9:30
  • $\begingroup$ Don't be lazy dude you have to brainstorm a bite $\endgroup$ – Guy Fsone Jan 20 '18 at 9:37
  • $\begingroup$ I mean, of course I can go through the proof, it's not about lazyness. My question was just if anyone knew a reference, that's it. If nobody knows it, I will braimstorm, no doubt. People write papers about optimal constants, so I figured that maybe it would be out of reach by me. $\endgroup$ – popoolmica Jan 20 '18 at 9:50

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