Geodesic
A Priori Lower Bound for the Minimal Eigenvalue of a Sturm–Liouville Problem with Boundary Conditions of the Second Type
Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 832-840 Cet article a éte moissonné depuis la source Math-Net.Ru

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We establish the attainability of the infimum $m_\gamma$ for the minimal eigenvalues of the boundary-value problems \begin{gather*} -y''+qy=\lambda y, \\ y'(0)=y'(1)=0 \end{gather*} as the nonnegative potential $q\in L_1[0,1]$ ranges over the unit sphere of the space $L_\gamma[0,1]$, where $\gamma\in (0,1)$. We also establish that, for $\gamma\leqslant 1-2\pi^{-2}$, the equality $m_\gamma=1$ holds and that, otherwise, the inequality $m_\gamma<1$ is valid.
Mots-clés : Sturm–Liouville problem, infimum, Hölder's inequality
Keywords: Lagrange finite-increment theorem, minimal eigenvalue, the space $L_\gamma[0,1]$, $\gamma\in (0,1)$. 
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     title = {A {Priori} {Lower} {Bound} for the {Minimal} {Eigenvalue} of a {Sturm{\textendash}Liouville} {Problem} with {Boundary} {Conditions} of the {Second} {Type}},
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A. A. Vladimirov; E. S. Karulina. A Priori Lower Bound for the Minimal Eigenvalue of a Sturm–Liouville Problem with Boundary Conditions of the Second Type. Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 832-840. http://geodesic.mathdoc.fr/item/MZM_2015_97_6_a2/

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