A Priori Lower Bound for the Minimal Eigenvalue of a Sturm--Liouville Problem with Boundary Conditions of the Second Type

A. A. Vladimirov ; E. S. Karulina

Matematičeskie zametki, Tome 97 (2015) no. 6, pp. 832-840

Résumé

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 $m_\gamma1$ is valid.

Détail

Mots-clés : Sturm–Liouville problem, infimum
Keywords: Lagrange finite-increment theorem, minimal eigenvalue, Hölder's inequality, the space $L_\gamma[0,1]$, $\gamma\in (0,1)$.