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

There are considered classical solutions nu of the Stokes system in the ball B = {x is an element of R-n : vertical bar x vertical bar < 1}, which are continuous up to the boundary. We derive the- optimal constant c = c(pi) such that, for all x is an element of B, vertical bar nu(x)vertical bar <= c(xi is an element of delta B)max vertical bar nu(xi)vertical bar (*) holds for all such functions. We show that c(n) = max(x is an element of B) c(n) (x) exists, where c(n) (x) is the minimal constant in (*) for any fixed x is an element of B. The constants c,( z) 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 -> partial derivative B and as n -> infinity 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.