Geodesic
Sharp estimates for derivatives of functions in the Sobolev classes $\mathring W_2^r(-1,1)$
Informatics and Automation, Function theory and differential equations, Tome 269 (2010), pp. 143-149

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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 $\sqrt2(2r)!(1+O(1/r))$, $r\to\infty$. 
@article{TRSPY_2010_269_a10,
     author = {G. A. Kalyabin},
     title = {Sharp estimates for derivatives of functions in the {Sobolev} classes $\mathring W_2^r(-1,1)$},
     journal = {Informatics and Automation},
     pages = {143--149},
     publisher = {mathdoc},
     volume = {269},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_269_a10/}
}
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G. A. Kalyabin. Sharp estimates for derivatives of functions in the Sobolev classes $\mathring W_2^r(-1,1)$. Informatics and Automation, Function theory and differential equations, Tome 269 (2010), pp. 143-149. http://geodesic.mathdoc.fr/item/TRSPY_2010_269_a10/