Spectral upper bound for the torsion function of symmetric stable processes
HTML articles powered by AMS MathViewer
- by Hugo Panzo PDF
- Proc. Amer. Math. Soc. 150 (2022), 1241-1255 Request permission
Abstract:
We prove a spectral upper bound for the torsion function of symmetric stable processes that holds for convex domains in $\mathbb {R}^d$. Our bound is explicit and captures the correct order of growth in $d$, improving upon the existing results of Giorgi and Smits [Indiana Univ. Math. J. 59 (2010), pp. 987–1011] and Biswas and Lőrinczi [J. Differential Equations 267 (2019), pp. 267–306]. Along the way, we make progress towards a torsion analogue of Chen and Song’s [J. Funct. Anal. 226 (2005), pp. 90–113] two-sided eigenvalue estimates for subordinate Brownian motion.References
- Rodrigo Bañuelos and Tom Carroll, Brownian motion and the fundamental frequency of a drum, Duke Math. J. 75 (1994), no. 3, 575–602. MR 1291697, DOI 10.1215/S0012-7094-94-07517-0
- R. Bañuelos, P. Mariano, and J. Wang, Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian, arXiv:2003.06867, 2020.
- M. van den Berg, Spectral bounds for the torsion function, Integral Equations Operator Theory 88 (2017), no. 3, 387–400. MR 3682197, DOI 10.1007/s00020-017-2371-0
- M. van den Berg and G. Buttazzo, On capacity and torsional rigidity, Bull. Lond. Math. Soc. 53 (2021), no. 2, 347–359. MR 4239178, DOI 10.1112/blms.12422
- M. van den Berg and Tom Carroll, Hardy inequality and $L^p$ estimates for the torsion function, Bull. Lond. Math. Soc. 41 (2009), no. 6, 980–986. MR 2575328, DOI 10.1112/blms/bdp075
- Michiel van den Berg, Estimates for the torsion function and Sobolev constants, Potential Anal. 36 (2012), no. 4, 607–616. MR 2904636, DOI 10.1007/s11118-011-9246-9
- Michiel van den Berg, Giuseppe Buttazzo, and Aldo Pratelli, On relations between principal eigenvalue and torsional rigidity, Commun. Contemp. Math. 23 (2021), no. 8, Paper No. 2050093, 28. MR 4348945, DOI 10.1142/S0219199720500935
- Jean Bertoin, Subordinators: examples and applications, Lectures on probability theory and statistics (Saint-Flour, 1997) Lecture Notes in Math., vol. 1717, Springer, Berlin, 1999, pp. 1–91. MR 1746300, DOI 10.1007/978-3-540-48115-7_{1}
- Anup Biswas and József Lőrinczi, Universal constraints on the location of extrema of eigenfunctions of non-local Schrödinger operators, J. Differential Equations 267 (2019), no. 1, 267–306. MR 3944272, DOI 10.1016/j.jde.2019.01.007
- Krzysztof Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), no. 1, 43–80. MR 1438304, DOI 10.4064/sm-123-1-43-80
- Krzysztof Bogdan, Tomasz Byczkowski, Tadeusz Kulczycki, Michal Ryznar, Renming Song, and Zoran Vondraček, Potential analysis of stable processes and its extensions, Lecture Notes in Mathematics, vol. 1980, Springer-Verlag, Berlin, 2009. Edited by Piotr Graczyk and Andrzej Stos. MR 2569321, DOI 10.1007/978-3-642-02141-1
- D. L. Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26 (1977), no. 2, 182–205. MR 474525, DOI 10.1016/0001-8708(77)90029-9
- Zhen-Qing Chen and Renming Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal. 226 (2005), no. 1, 90–113. MR 2158176, DOI 10.1016/j.jfa.2005.05.004
- Bartłomiej Dyda, Alexey Kuznetsov, and Mateusz Kwaśnicki, Eigenvalues of the fractional Laplace operator in the unit ball, J. Lond. Math. Soc. (2) 95 (2017), no. 2, 500–518. MR 3656279, DOI 10.1112/jlms.12024
- Marcel Filoche and Svitlana Mayboroda, Universal mechanism for Anderson and weak localization, Proc. Natl. Acad. Sci. USA 109 (2012), no. 37, 14761–14766. MR 2990982, DOI 10.1073/pnas.1120432109
- Rupert L. Frank, Eigenvalue bounds for the fractional Laplacian: a review, Recent developments in nonlocal theory, De Gruyter, Berlin, 2018, pp. 210–235. MR 3824213, DOI 10.1515/9783110571561-007
- R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc. 101 (1961), 75–90. MR 137148, DOI 10.1090/S0002-9947-1961-0137148-5
- Tiziana Giorgi and Robert G. Smits, Principal eigenvalue estimates via the supremum of torsion, Indiana Univ. Math. J. 59 (2010), no. 3, 987–1011. MR 2779069, DOI 10.1512/iumj.2010.59.3935
- Antoine Henrot, Ilaria Lucardesi, and Gérard Philippin, On two functionals involving the maximum of the torsion function, ESAIM Control Optim. Calc. Var. 24 (2018), no. 4, 1585–1604. MR 3922448, DOI 10.1051/cocv/2017069
- T. R. Hurd and A. Kuznetsov, On the first passage time for Brownian motion subordinated by a Lévy process, J. Appl. Probab. 46 (2009), no. 1, 181–198. MR 2508513, DOI 10.1239/jap/1238592124
- Mateusz Kwaśnicki, Fractional Laplace operator and its properties, Handbook of fractional calculus with applications. Vol. 1, De Gruyter, Berlin, 2019, pp. 159–193. MR 3888401
- Lee Lorch, Some inequalities for the first positive zeros of Bessel functions, SIAM J. Math. Anal. 24 (1993), no. 3, 814–823. MR 1215440, DOI 10.1137/0524050
- Jianfeng Lu and Stefan Steinerberger, A dimension-free Hermite-Hadamard inequality via gradient estimates for the torsion function, Proc. Amer. Math. Soc. 148 (2020), no. 2, 673–679. MR 4052204, DOI 10.1090/proc/14843
- Phanuel Mariano and Hugo Panzo, Conformal Skorokhod embeddings and related extremal problems, Electron. Commun. Probab. 25 (2020), Paper No. 42, 11. MR 4112773, DOI 10.1214/20-ecp324
- L. E. Payne, Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 3-4, 251–265. MR 616778, DOI 10.1017/S0308210500020102
- George Pólya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quart. Appl. Math. 6 (1948), 267–277. MR 26817, DOI 10.1090/S0033-569X-1948-26817-9
- Renming Song and Zoran Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Related Fields 125 (2003), no. 4, 578–592. MR 1974415, DOI 10.1007/s00440-002-0251-1
- Renming Song and Zoran Vondraček, On the relationship between subordinate killed and killed subordinate processes, Electron. Commun. Probab. 13 (2008), 325–336. MR 2415141, DOI 10.1214/ECP.v13-1388
- Alain-Sol Sznitman, Brownian motion, obstacles and random media, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1717054, DOI 10.1007/978-3-662-11281-6
- Francesco Tricomi, Sulle funzioni di Bellel di ordine e argomento pressochè uguali, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 83 (1949), 3–20 (Italian). MR 34478
- Hendrik Vogt, $L_\infty$-estimates for the torsion function and $L_\infty$-growth of semigroups satisfying Gaussian bounds, Potential Anal. 51 (2019), no. 1, 37–47. MR 3981441, DOI 10.1007/s11118-018-9701-y
- J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), 563–564. MR 29448, DOI 10.2307/2304460
Additional Information
- Hugo Panzo
- Affiliation: Technion – Israel Institute of Technology, Haifa 32000, Israel
- MR Author ID: 1003446
- ORCID: 0000-0003-4392-672X
- Email: panzo@campus.technion.ac.il
- Received by editor(s): July 19, 2020
- Received by editor(s) in revised form: March 25, 2021, and June 21, 2021
- Published electronically: January 5, 2022
- Additional Notes: This work was supported at the Technion by a Zuckerman Fellowship
- Communicated by: Zhen-Qing Chen
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 1241-1255
- MSC (2020): Primary 35P15, 60G52; Secondary 35R11, 60J45, 60J65
- DOI: https://doi.org/10.1090/proc/15764
- MathSciNet review: 4375718