Minimization of the zeroth Neumann eigenvalues with integrable potentials
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 501-523.

Soit λ 0 (q) 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 𝐋 ˜ 1 (r)= inf q λ 0 (q) pour les potentiels q de valeur moyenne zéro et de norme L 1 égale à r. Le résultat est 𝐋 ˜ 1 (r)=-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 λ 0 (q) be the zeroth Neumann eigenvalue of the Sturm–Liouville operator with the potential q. In this paper we will solve the minimization problem 𝐋 ˜ 1 (r)= inf q λ 0 (q), where potentials q have mean value zero and L 1 norm r. The final result is 𝐋 ˜ 1 (r)=-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.

DOI : 10.1016/j.anihpc.2012.01.007
Classification : 34L15, 34L40, 49R05
Mots clés : Eigenvalue, Potential, Extremal value, Measure differential equation, Weak topology
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Zhang, Meirong. Minimization of the zeroth Neumann eigenvalues with integrable potentials. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 4, pp. 501-523. doi : 10.1016/j.anihpc.2012.01.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.01.007/

[1] C. Bandle, Extremal problems for eigenvalues of the Sturm–Liouville type, General Inequalities 5, Oberwolfach, 1986, Internat. Schriftenreihe Numer. Math. vol. 80, Birkhäuser, Basel (1987), 319-336 | MR

[2] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model, I. Species persistence, J. Math. Biol. 51 (2005), 75-113 | MR | Zbl

[3] M. Carter, B. Van Brunt, The Lebesgue–Stieltjes Integral: A Practical Introduction, Springer-Verlag, New York (2000) | MR | Zbl

[4] A. Derlet, J.-P. Gossez, P. Takáč, Minimization of eigenvalues for a quasilinear elliptic Neumann problem with indefinite weight, J. Math. Anal. Appl. 371 (2010), 69-79 | MR | Zbl

[5] N. Dunford, J.T. Schwartz, Linear Operators, Part I, Interscience Publishers, New York (1958) | MR | Zbl

[6] J. Fernández Bonder, E. Lami Dozo, J. Rossi, Symmetry properties for the extremals of the Sobolev trace embedding, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 795-805 | EuDML | Numdam | MR | Zbl

[7] S. Karaa, Extremal eigenvalues and their associated nonlinear equations, Boll. Un. Mat. Ital. B (7) 10 (1996), 625-649 | MR | Zbl

[8] S. Karaa, Sharp estimates for the eigenvalues of some differential equations, SIAM J. Math. Anal. 29 (1998), 1279-1300 | MR | Zbl

[9] M.G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, Amer. Math. Soc. Transl. Ser. 2 no. 1 (1955), 163-187 | MR | Zbl

[10] Y. Lou, E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math. 23 (2006), 275-292 | MR | Zbl

[11] W. Magnus, S. Winkler, Hillʼs Equation, Dover, New York (1979) | MR

[12] T.J. Mahar, B.E. Willner, An extremal eigenvalue problem, Comm. Pure Appl. Math. 29 (1976), 517-529 | MR | Zbl

[13] R.E. Megginson, An Introduction to Banach Space Theory, Graduate Texts Math. vol. 183, Springer-Verlag, New York (1998) | MR | Zbl

[14] G. Meng, Continuity of solutions and eigenvalues in measures with weak⁎ topology, Doctoral Dissertation, Tsinghua University, Beijing, 2009.

[15] G. Meng, P. Yan, M. Zhang, Spectrum of one-dimensional p-Laplacian with an indefinite integrable weight, Mediterr. J. Math. 7 (2010), 225-248 | MR | Zbl

[16] A.B. Mingarelli, Volterra–Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Lect. Notes Math. vol. 989, Springer-Verlag, Berlin (1983) | Zbl

[17] M. Möller, A. Zettl, Differentiable dependence of eigenvalues of operators in Banach spaces, J. Operator Theory 36 (1996), 335-355 | MR | Zbl

[18] L. Notarantonio, Extremal properties of the first eigenvalue of Schrödinger-type operators, J. Funct. Anal. 156 (1998), 333-346 | MR | Zbl

[19] J. Pöschel, E. Trubowitz, The Inverse Spectral Theory, Academic Press, New York (1987) | MR | Zbl

[20] L. Roques, F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci. 210 (2007), 34-59 | MR | Zbl

[21] Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore (1992) | MR | Zbl

[22] Q. Wei, G. Meng, M. Zhang, Extremal values of eigenvalues of Sturm–Liouville operators with potentials in L 1 balls, J. Differential Equations 247 (2009), 364-400 | MR | Zbl

[23] P. Yan, M. Zhang, Continuity in weak topology and extremal problems of eigenvalues of the p-Laplacian, Trans. Amer. Math. Soc. 363 (2011), 2003-2028 | MR | Zbl

[24] D.P. Zeragiya, Variational problems associated with extremal properties of the first eigenvalue of a problem of Sturm–Liouville type, Differential Equations 32 (1996), 1331-1336 | MR | Zbl

[25] A. Zettl, Sturm–Liouville Theory, Math. Surveys & Monographs vol. 121, Amer. Math. Soc., Providence, RI (2005) | MR | Zbl

[26] M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc. (2) 64 (2001), 125-143 | MR | Zbl

[27] M. Zhang, Certain classes of potentials for p-Laplacian to be non-degenerate, Math. Nachr. 278 (2005), 1823-1836 | MR | Zbl

[28] M. Zhang, Continuity in weak topology: Higher order linear systems of ODE, Sci. China Ser. A 51 (2008), 1036-1058 | MR | Zbl

[29] M. Zhang, Extremal values of smallest eigenvalues of Hillʼs operators with potentials in L 1 balls, J. Differential Equations 246 (2009), 4188-4220 | MR | Zbl

[30] M. Zhang, Extremal eigenvalues of measure differential equations with fixed variation, Sci. China Math. 53 (2010), 2573-2588 | MR | Zbl

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