Minimization of the zeroth Neumann eigenvalues with integrable potentials

Zhang, Meirong

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 501-523.

Soit ${\lambda }_0\left(q\right)$ la zéro-ème valeur propre de Neumann de lʼopérateur de Sturm–Liouville pour un potentiel intégrable q de lʼintervalle $[0,1]$. Dans cet article nous résolvons le problème de minimisation ${\widetilde{𝐋}}_1\left(r\right)={\inf }_q{\lambda }_0\left(q\right)$ pour les potentiels q de valeur moyenne zéro et de norme $L^1$ égale à r. Le résultat est ${\widetilde{𝐋}}_1\left(r\right)=-r^2/4$. Lʼapproche est une combinaison de méthode variationnelle et de procédé de limite, utilisant des résultats de continuité des solutions et des valeurs propres dʼéquations linéaires en les potentiels et les mesures dans des topologies faibles. Ces valeurs extrémales peuvent donner des estimations optimales sur les zéro-èmes valeurs propres de Neumann.

For an integrable potential q on the unit interval, let ${\lambda }_0\left(q\right)$ be the zeroth Neumann eigenvalue of the Sturm–Liouville operator with the potential q. In this paper we will solve the minimization problem ${\widetilde{𝐋}}_1\left(r\right)={\inf }_q{\lambda }_0\left(q\right)$, where potentials q have mean value zero and $L^1$ norm r. The final result is ${\widetilde{𝐋}}_1\left(r\right)=-r^2/4$. The approach is a combination of variational method and limiting process, with the help of continuity results of solutions and eigenvalues of linear equations in potentials and in measures with weak topologies. These extremal values can yield optimal estimates on the zeroth Neumann eigenvalues.

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