I am reading a paper about Navier-Stokes Equations, in which the author gives the inequality bellow
$$\|u\|_{L^2(O_R)}\le CR^{\frac{1}{2}}\|\nabla u\|_{L^2(O_R)}$$
where $u$ is a $3D$ vector field, $O_R=(B_R(0)-B_{R-1}(0))\times(0,1)\subset\mathbb{R}^3, R>0$, $C>0$ is a constant which is independent of $R$ .
I have check all kinds of Poincaré's inequalities, and I cant find one with exact value of the constant, whether additional conditions are required?
Or, under what circumstances can this constant be determined
2026-03-28 21:34:17.1774733657
How to determine the exact value of the constant in Poincaré's inequality?
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