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.
It's here where I will trail off and let you go discover the rest of the details on your own. If you get stuck you can either look in Strauss's book Partial Differential Equations: An Introduction, or you can hit up google. For instance, I think this link has what you're looking for. The key point is that you will find a connection with all sorts of beautiful classical mathematics related to Bessel functions and their zeroes (I can't help but throw out a reference to the tour-de-force by Watson, A Treatise on the Theory of Bessel Functions).
Let me know if you'd like more help or extra references. I'm happy to help!