Sharp estimates for derivatives of functions in the Sobolev classes W̊2 r(-1, 1)

Explicit formulas are obtained for the maximum possible values of the derivatives f(k)(x), x ∈ (-1, 1), k ∈ {0,1,...,r - 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r - 1 at the points ±1 and are such that {double pipe}f(r){double pipe}L2(-1,1) ≤ 1. As a corollary, it is shown that the first eigenvalue λ1,r of the operator (-D2)r with these boundary conditions is √2 (2r)! (1 + O(1/r)), r → ∞. © 2010 Pleiades Publishing, Ltd.