# The best constant of Poincare inequality can be determined by eigenvalue of Laplace operator

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!

• Searching "Rayleigh quotient, eigenfunction, Laplacian" brings up quite a few results, such as these lecture notes with a proof of the above, and a more general result. – user147263 Jan 2 '15 at 5:06
• @Behaviour Thank you for reference. I'll check it out! – spatially Jan 2 '15 at 13:59