Sharp estimates for derivatives of functions in the Sobolev classes $\mathring W_2^r(-1,1)$

Explicit formulas are obtained for the maximum possible values of the derivatives $f^{(k)}(x)$, $x\in(-1,1)$, $k\in\{0,1,\dots,r-1\}$, for functions $f$ that vanish together with their (absolutely continuous) derivatives of order up to $\le r-1$ at the points $\pm1$ and are such that $\|f^{(r)}\|_{L_2(-1,1)}\le1$. As a corollary, it is shown that the first eigenvalue $\lambda_{1,r}$ of the operator $(-D^2)^r$ with these boundary conditions is $\sqrt2(2r)!(1+O(1/r))$, $r\to\infty$.