# Poincaré constant for a ball (circle)

I've been recently looking for a best possible Poincaré constant for a particular domains $\Omega$ (it's related to my previous question Unique weak solution to Helmholtz equation on a square) for $W_0^{1,2}(\Omega)$-functions.

If $\Omega = (0,1)^2$, than I was able able to prove that \begin{equation*} ||u||_2^2\leq \frac{1}{16}||\nabla u||_2^2,\quad\forall u\in W_0^{1,2}((0,1)^2). \end{equation*}

I was wondering what happens if $\Omega=B(0, 1)\subseteq\mathbb{R}^2$? I can circumsribe a square about the circle and I get that \begin{equation*} ||u||_2^2\leq \frac{1}{4}||\nabla u||_2^2,\quad\forall u\in W_0^{1,2}(B(0,1)), \end{equation*} but I believe there must be a clever way how to do it. I don't necessarily need the best possible constant (i.e. the smallest one), but I'm looking for some reasonable estimate (without some magic tricks) that gives a constant that is as small as possible.

Thank you for any help:)

The key to what you're looking for is Rayleigh's theorem, which states that $$\lambda_1 = \min_{0 \neq u \in W^{1,2}_0(\Omega) } \frac{\int_\Omega |\nabla u|^2 } {\int_\Omega u^2},$$ where $\lambda_1 >0$ is the principal eigenvalue of the Laplacian with Dirichlet boundary condition, i.e. $$\begin{cases} -\Delta u = \lambda_1 u &\text{in } \Omega \\ u = 0 & \text{on }\partial \Omega. \end{cases}$$ Why is this useful? For any $u \in W^{1,2}_0(\Omega)$ such that $u \neq 0$ the Rayleigh theorem tells us that $$\lambda_1 \int_\Omega u^2 \le \int_\Omega |\nabla u|^2,$$ but this inequality also trivially holds for $u=0$ and thus for all $u \in W^{1,2}_0(\Omega)$. Thus $1/\lambda_1$ is the best constant in the Poincaré inequality since the infimum is achieved by the solution to the Dirichlet problem. Now, the crucial feature of this is that for a ball, namely $\Omega = B(0,r)$, we can explicitly compute the eigenfunctions and eigenvalues of the Laplacian by using the classical PDE technique of separation of variables.