Given $\Omega\subset \mathbb R^N$ be open bounded and smooth boundary. Then for $u\in H_0^1(\Omega)$ we have well-known Poincare inequality $$ \int_\Omega \lvert u\lvert^2dx\leq C\int_\Omega \lvert\nabla u\lvert^2dx \tag 1$$
I'm interested in finding the best constant $C$ here. Re-write $(1)$, we have $$ \frac{1}{C}\leq \frac{\int_\Omega \lvert\nabla u\lvert^2dx}{ \int_\Omega \lvert u\lvert^2dx} $$ and hence we only need to find $$ \inf_{u\in H_0^1(\Omega)} \int_\Omega \lvert\nabla u\lvert^2dx:=\alpha$$ over the admissible set $M:=\{u\in H_0^1(\Omega)\,\|u\|_{L^2(\Omega)}=1\}$. Then by Rayleigh Quotient theorem we have $\alpha=\lambda_1$ where $\lambda_1$ is the first eigenvalue of laplace operator $-\Delta $.
Hence the best constant of Poincare inequality is just $1/\lambda_1$? Am I correct?
I think this problem has been well studied. So if you know where I can find a good reference, please kindly direct me there. Thank you!