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 $c_n = \mathrm {max}_{x \in \partial B} c_n (x)$ exists, where $c_n(x)$ is the minimal constant in (*) 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.