Consider Poincare inequality in the form of: $|u|_{0,\Omega} \leq C|u|_{1,\Omega} \text{ for every } u\in H^1_0(\Omega),$ where $\Omega \subset \mathbb{R}^n$ is a bounded open set. Find the best possible value of $C$.
Any type of help will be appriciated. Thanks in advance.
Finding the best constant for Poincare inequality (or korn's inequality) is a long standing problem. Unfortunately, there is no general answer. (not I am known of). However, for some specially domains, there is something you can do.
For example, if $\Omega$ is a ball, then the best constant is the radius of the ball(or something similar). I would suggest you to read this book, chapter 12.2. Inside you could find more informations, for example ,the poincare for star-sharped domains, and more.