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!