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

### Werner Kratz

Universität Ulm, Germany

## Abstract

There are considered classical solutions $ν$ of the Stokes system in the ball $B={Ix∈R_{n}:∣x∣<1}$, which are continuous up to the boundary. We derive the-optimal constant $c=c_{n}$, such that, for all $x∈B$,

holds for all such functions. We show that $c_{n}=max_{x∈∂B}c_{n}(x)$ exists, where $c_{n}(x)$ is the minimal constant in (*) for any fixed $x∈B$. 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 $x→∂B$ and as $n→∞$ 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.

## Cite this article

Werner Kratz, An Extremal Problem Related to the Maximum Modulus Theorem for Stokes Functions. Z. Anal. Anwend. 17 (1998), no. 3, pp. 599–613

DOI 10.4171/ZAA/841