Geodesic
Majorants for eigenvalues of Sturm-Liouville problems with potentials lying in balls of weighted spaces
Sbornik. Mathematics, Tome 208 (2017) no. 9, pp. 1298-1311 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the problem concerning an exact a priori majorant for the least eigenvalue of the Sturm-Liouville problem $$ -y''+qy=\lambda y,\qquad y(0)=y(1)=0 $$ with a condition of the form $\displaystyle\int_0^1 rq^\gamma\,dx\leqslant 1$ on the potential, where the weight $r\in C(0,1)$ is uniformly positive on the interval $(0,1)$. We give a constructive proof that this majorant is attainable for all $\gamma>1$ and, for a certain natural extension of the class of admissible potentials, also for $\gamma=1$. Bibliography: 9 titles.
Keywords: eigenvalue, Sobolev space.
Mots-clés : Sturm-Liouville problem 
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     title = {Majorants for eigenvalues of {Sturm-Liouville} problems with potentials lying in balls of weighted spaces},
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A. A. Vladimirov. Majorants for eigenvalues of Sturm-Liouville problems with potentials lying in balls of weighted spaces. Sbornik. Mathematics, Tome 208 (2017) no. 9, pp. 1298-1311. http://geodesic.mathdoc.fr/item/SM_2017_208_9_a2/

[1] Yu. V. Egorov, V. A. Kondrat'ev, “On estimates of the first eigenvalue of the Sturm–Liouville problem”, Russian Math. Surveys, 39:2 (1984), 149–150 | DOI | MR | Zbl

[2] V. A. Vinokurov, V. A. Sadovnichii, “On the range of variation of an eigenvalue when the potential is varied”, Dokl. Math., 68:2 (2003), 247–252 | MR | Zbl

[3] S. S. Ezhak, “Otsenki pervogo sobstvennogo znacheniya zadachi Shturma–Liuvillya s usloviyami Dirikhle”, Kachestvennye svoistva reshenii differentsialnykh uravnenii i smezhnye voprosy spektralnogo analiza, ed. I. V. Astashova, Yuniti-Dana, M., 2012, 517–559

[4] M. Yu. Telnova, “Otsenki pervogo sobstvennogo znacheniya zadachi Shturma–Liuvillya s usloviyami Dirikhle i vesovym integralnym usloviem”, Kachestvennye svoistva reshenii differentsialnykh uravnenii i smezhnye voprosy spektralnogo analiza, ed. I. V. Astashova, Yuniti-Dana, M., 2012, 608–647

[5] M. Yu. Telnova, “Ob otsenkakh sverkhu pervogo sobstvennogo znacheniya zadachi Shturma–Liuvillya s vesovym integralnym usloviem”, Vestn. SamGU. Estestvennonauchn. ser., 2015, no. 6(128), 124–129

[6] A. A. Markov, “On constructive mathematics”, Amer. Math. Soc. Transl. Ser. 2, 98, Amer. Math. Soc., Providence, RI, 1971, 1–9 | DOI | MR | Zbl

[7] E. S. Karulina, A. A. Vladimirov, “The Sturm–Liouville problem with singular potential and the extrema of the first eigenvalue”, Tatra Mt. Math. Publ., 54 (2013), 101–118 | MR | Zbl

[8] M. I. Neiman-Zade, A. A. Shkalikov, “Schrödinger operators with singular potentials from spaces of multiplicators”, Math. Notes, 66:5 (1999), 599–607 | DOI | DOI | MR | Zbl

[9] F. Riesz, B. Sz.-Nagy, Leçons d'analyse fonctionnelle, 4ème éd., Gauthier-Villars, Paris; Akadémiai Kiadó, Budapest, 1965, viii+490 pp. | MR | MR | Zbl