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Zeitschrift für Analysis und ihre Anwendungen

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Volume 17, Issue 3, 1998, pp. 599–613
DOI: 10.4171/ZAA/841

Published online: 1998-09-30

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

Werner Kratz[1]

(1) Universität Ulm, Germany

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 $x \in B$, $$| \nu (x) | ≤ c \ \mathrm {max}_{\xi \in \partial B} | \nu(\xi)| \ \ \ (*)$$ 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.

Keywords: Stokes system, maximum modulus theorem, Stokes-Poisson integral formula, norm of linear mappings

Kratz Werner: An Extremal Problem Related to the Maximum Modulus Theorem for Stokes Functions. Z. Anal. Anwend. 17 (1998), 599-613. doi: 10.4171/ZAA/841