An Extremal Problem Related to the Maximum Modulus Theorem for Stokes Functions

Abstract

There are considered classical solutions $\nu$ of the Stokes system in the ball $B=\{Ix\in {\mathbb{R}}^n:|x|<1\}$, which are continuous up to the boundary. We derive the-optimal constant $c=c_n$, such that, for all $x\in B$,

holds for all such functions. We show that $c_n={\max }_{x\in \partial B}{c}_n(x)$ exists, where $c_n(x)$ is the minimal constant in ($\ast$) for any fixed $x\in B$. The constants $c_n(x)$ are determined explicitly via the Stokes–Poisson integral formula and via a general theorem on the norm of certain linear mappings given by some matrix kernel. Moreover, the asymptotic behaviour of the $c_n(x)$ as $x\to \partial B$ and as $n\to \infty$ is derived.
In the concluding section the general result on the norm of linear mappings is used to prove two inequalities: one for linear combinations of Fourier coefficients and the other from matrix analysis.