JournalszaaVol. 17, No. 3pp. 599–613

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

  • Werner Kratz

    Universität Ulm, Germany
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Abstract

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

ν(x)c maxξBν(ξ)   ()| \nu (x) | ≤ c \ \mathrm {max}_{\xi \in \partial B} | \nu(\xi)| \ \ \ (*)

holds for all such functions. We show that cn=maxxBcn(x)c_n = \mathrm {max}_{x \in \partial B} c_n (x) exists, where cn(x)c_n(x) is the minimal constant in (*) for any fixed xBx \in B. The constants cn(x)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 cn(x)c_n(x) as xBx \to \partial B and as nn \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.

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