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On Courant and Pleijel theorems for sub-Riemannian Laplacians (2402.13953v3)

Published 21 Feb 2024 in math.SP and math.AP

Abstract: We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel's theorem, which states that, as soon as the dimension is strictly larger than 1, the number of nodal domains of an eigenfunction corresponding to the k-th eigenvalue is strictly (and uniformly, in a certain sense) smaller than k for large k. In the first part of this paper we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups. In the second part, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way we improve known bounds on the optimal constants in the Faber-Krahn and isoperimetric inequalities on these groups.

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References (89)
  1. M. Abramowitz, I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, No. 55 U. S. Government Printing Office, Washington, D.C., 1964
  2. C. Anné. Bornes sur la multiplicité. Prépublications EPFL (1992).
  3. Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3. arXiv:2306.10757v1 (2023).
  4. T. Aubin. Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geometry 11 (1976), no. 4, 573–598.
  5. H. Bahouri. Sur la propriété de prolongement unique pour les opérateurs de Hörmander. Journées équations aux dérivées partielles (1983), p. 1-7 and Annales de l’Institut Fourier, tome 36, no 4 (1986), 137–155.
  6. Non-analytic-hypoellipticity for some degenerate elliptic operator. Bulletin of the American Mathematical Society, Volume 78, Number 3, 483–486, (1972).
  7. Sobolev inequalities in disguise. Indiana Univ. Math. J. 44 (1995), no. 4, 1033–1074.
  8. Sub-Riemannian Geometry. Birkhäuser (1996).
  9. Inégalités isopérimétriques et applications. Annales scientifiques de l’École Normale Supérieure, Série 4, Tome 15, no 3, 513–541 (1982).
  10. J.M. Bony. Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour des opérateurs elliptiques dégénérés. Annales de l’institut Fourier, (1969), Vol. 19, no 1, 277–304.
  11. Moser–Trudinger and Beckner–Onofri’s inequalities on the CR sphere. Ann. of Math. (2) 177 (2013), no. 1, 1–52.
  12. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Progr. Math., 259. Birkhäuser Verlag, Basel,( 2007).
  13. G. Carron. Inégalités isopérimétriques de Faber-Krahn et conséquences. Sémin. Congr., 1, Société Mathématique de France, Paris, 1996, 205–232.
  14. L. G. Chambers. An upper bound for the first zero of Bessel functions. Math. Comp., 38(158), (1982), 589–591.
  15. Umbilicity and characterization of Pansu spheres in the Heisenberg group. J. Reine Angew. Math. 738 (2018), 203–235.
  16. A Codazzi-like equation and the singular set for C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT smooth surfaces in the Heisenberg group. J. Reine Angew. Math. 671 (2012), 131–198.
  17. Weyl’s law for singular Riemannian manifolds. Journal de Mathématiques Pures et Appliquées Volume 181, January 2024, 113–151
  18. W. S. Cohn, G. Lu. Best constants for Moser-Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J. 50 (2001), no. 4, 1567–1591.
  19. Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case. Duke Math. Journal, 167 (1): 109–174 (2018).
  20. Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results. Annales Henri Lebesgue 4 (2021): 897–971.
  21. Spectral asymptotics for sub-Riemannian Laplacians. ArXiv 2022.
  22. Methods of Mathematical Physics: Partial Differential Equations, Vol. 2. John Wiley and Sons, (2008).
  23. Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot–Carathéodory spaces. Memoirs of the AMS, Vol. 182, No 537 (2006).
  24. A partial solution of the isoperimetric problem for the Heisenberg group. Forum Math. 20 (2008), no. 1, 99–143.
  25. M. Derridj. Un problème aux limites pour une classe d’opérateurs du second ordre hypoelliptiques. Ann. Inst. Fourier, Grenoble, 21, 4 (1971), 99–148.
  26. Pleijel nodal domain theorem in non-smooth setting. ArXiv:2307.13983 (2023).
  27. On the square of the zeros of Bessel functions. SIAM J. Math. Anal. 15 (1984), no. 1, 206–212.
  28. Nodal sets of Eigenfunctions of sub-Laplacians. ArXiv January 2023. International Mathematics Research Notices (2023), to appear.
  29. G. Folland. A fundamental solution for a subelliptic operator. Bull. Amer. Math. Soc. 79 (1973), 373–376.
  30. G. Folland. Subelliptic estimates and function spaces on nilpotent groups. Ark. Math. 13 (1975), 161–207.
  31. Estimates for the ∂¯bsubscript¯𝑏\overline{\partial}_{b}over¯ start_ARG ∂ end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT-complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974), 429–522.
  32. Meyers–Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math. 22 (1996), no. 4, 859–890.
  33. Schrödinger operators: eigenvalues and Lieb–Thirring inequalities. Cambridge Stud. Adv. Math., 200, Cambridge University Press, Cambridge (2023).
  34. Rupert L. Frank and Elliott H. Lieb. Sharp constants in several inequalities on the Heisenberg group. Annals of mathematics (2012), 349–381.
  35. The nonlinear Schrödinger equation for orthonormal functions II: application to Lieb–Thirring inequalities. Commun. Math. Phys. 384 (2021), 1783–1828.
  36. N. Garofalo. Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Differential Equations 104 (1993), no. 1, 117–146.
  37. Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144.
  38. Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot–Caratheodory space. J. Anal. Math. 74 (1998), 67–97.
  39. R.W. Goodman. Nilpotent Lie groups. Lecture Notes in Mathematics. No 562. Springer (1976).
  40. Mem. Amer. Math. Soc. 145 (2000), no. 688.
  41. A. M. Hansson and A. Laptev. Sharp spectral inequalities for the Heisenberg Laplacian. London Math. Soc. Lecture Note Ser., 354, 100–115, (2008).
  42. On Pleijel’s nodal domain theorem for the Robin problem. arXiv: 2303.08094 (2023).
  43. B. Helffer. Conditions nécessaires d’hypoanalyticité pour des opéerateurs invariants à gauche sur un groupe nilpotent gradué. J. Diff. Eq. 44 (1982) 460–481.
  44. Nodal domains and spectral minimal partitions. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009).
  45. Approximation d’un système de champs de vecteurs et applications à l’hypoellipticité. Ark. Mat. 17(1-2): 237-254 (1979).
  46. Hypoellipticité Maximale pour des Opérateurs Polynômes de Champs de Vecteurs. Progress in Mathematics. Birkhäuser. 1985
  47. B. Helffer and M. Persson Sundqvist. On nodal domains in Euclidean balls. Proc. Amer.Math. Soc. 144 (11): 4777–4791 (2017).
  48. On the zeros of solutions of elliptic inequalities in bounded domains. J. Differential Equations 28 (1978), no.3, 345–353.
  49. L. Hörmander. Hypoelliptic second order differential equations. Acta Math. 119: 147-171 (1967).
  50. F. Jean. Control of nonholonomic systems: from sub-Riemannian geometry to motion planning. Monograph. Springer (2014).
  51. D. Jerison. The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. J. of Functional Analysis 43 (1981), Part I, 97–141, Part II, 224–257.
  52. Extremal for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. Journal of the AMS, Vol. 1, Number 1, January 1988.
  53. Maximal function methods for Sobolev spaces. Mathematical Surveys and Monographs, 257. American Mathematical Society, Providence, RI, 2021. xii+338 pp.
  54. Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184 (2000), no. 1, 87–111.
  55. C. Léna. Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions. Annales de l’Institut Fourier 69 (1), 283–301 (2019).
  56. G. P. Leonardi and S. Masnou. On the isoperimetric problem in the Heisenberg group ℍnsuperscriptℍ𝑛\mathbb{H}^{n}blackboard_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Ann. Mat. Pura Appl. (4) 184 (2005), no. 4, 533–553.
  57. G. P. Leonardi, S. Rigot. Isoperimetric sets on Carnot groups. Houston J. Math. 29 (2003), no. 3, 609–637.
  58. G. Leoni. A first course in Sobolev spaces. Second edition. Grad. Stud. Math., 181, American Mathematical Society, Providence, RI, (2017).
  59. Topics in spectral Geometry. AMS Graduate Studies in Mathematics series, volume 237 (2023).
  60. E. H. Lieb, M. Loss. Analysis Grad. Stud. Math., 14, American Mathematical Society, Providence, RI, 2001, xxii+346 pp.
  61. L. Lorch. Some inequalities for the first positive zeros of Bessel functions. SIAM J. Math. Anal. 24 (1993), no. 3, 814–823.
  62. F. Maggi. Sets of finite perimeter and geometric variational problems Cambridge Stud. Adv. Math., 135, Cambridge University Press, Cambridge, 2012, xx+454 pp.
  63. Formulas and Theorems for the special functions of mathematical Physics. Die Grundlehre der mathematischen Wissenschaften. Band 52. Third edition. Springer (1966).
  64. On the eigenvalues of a class of hypoelliptic operators. Math. Ann. 235 (1978), 55–85.
  65. On the eigenvalues of a class of hypoelliptic operators II. Global analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ. Calgary, Calgary, Alta., 1978), 201–247, Lecture Notes in Math., 755, Springer, Berlin, 1979.
  66. G. Métivier. Fonction spectrale et valeurs propres d’une classe d’opérateurs non elliptiques. Comm. in PDE 1 (1976), 467–519.
  67. G. Métivier. Hypoellipticité analytique sur des groupes nilpotents de rang 2222. Duke Math. Journal, Vol . 47, No 1 (1980), 195–221.
  68. John W. Milnor. Topology from the differentiable viewpoint. Princeton University Press, Princeton, NJ, (1997). Revised reprint of the 1965 original.
  69. A. Mohamed. Étude spectrale d’opérateurs hypoelliptiques á caractéristiques multiples I. Annales de l’Institut Fourier, Tome 32 (1982) no. 3, 39–90.
  70. R. Monti. Heisenberg isoperimetric problem. The axial case. Adv. Calc. Var. 1 (2008), no.1, 93–121.
  71. Non-tangentially accessible domains for vector fields. Indiana University Mathematics Journal, Vol. 54, No. 2 (2005).
  72. Convex isoperimetric sets in the Heisenberg group. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 2, 391–415.
  73. E. Müller-Pfeiffer. On the number of nodal domains for eigenfunctions of elliptic differential operators. J. London Math. Soc. (2) 31 (1985), no. 1, 91–100.
  74. B.D.S. Nagy. Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung. Acta Sci. Math. 10, 64–74 (1941).
  75. P. Pansu. An isoperimetric inequality for the Heisenberg group. Comptes Rendus Acad. Sc. Math. 295.2 (1982), 127–130.
  76. P. Pansu. An isoperimetric inequality on the Heisenberg group. Conference on differential geometry on homogeneous spaces (Torino, 1983). Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, 159–174 (1984).
  77. A. Pleijel. Remarks on Courant’s nodal theorem. Comm. Pure. Appl. Math., 9: 543–550 (1956).
  78. I. Polterovich. Pleijel’s nodal domain theorem for free membranes. Proc. Amer. Math. Soc. 137 (2009), no. 3, 1021–1024.
  79. M. Ritoré. A proof by calibration of an isoperimetric inequality in the Heisenberg group ℍnsuperscriptℍ𝑛\mathbb{H}^{n}blackboard_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 47–60.
  80. M. Ritoré, C. Rosales. Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍnsuperscriptℍ𝑛\mathbb{H}^{n}blackboard_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. J. Geom. Anal. 16 (2006), no. 4, 703–720.
  81. M. Ritoré, C. Rosales. Area-stationary surfaces in the Heisenberg group ℍ1superscriptℍ1\mathbb{H}^{1}blackboard_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT. Adv. Math. 219 (2008), no. 2, 633–671.
  82. E. Rodemich. The Sobolev inequality with best possible constant. Analysis Seminar Caltech, Spring 1966.
  83. L.P. Rothschild. A criterion for hypoellipticity of operators constructed of vector fields. Comm. in PDE 4 (6) (1979), 248–315.
  84. Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137, 248–315. (1976)
  85. G. Talenti. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.
  86. G. Talenti. The standard isoperimetric theorem. North-Holland Publishing Co., Amsterdam, (1993), 73–123.
  87. N. Th. Varopoulos. Analysis on nilpotent groups. Journal of Functional Analysis 66, 406–431 (1986).
  88. Convergence from power-law to logarithm-law in nonlinear scalar field equations. Arch. Ration. Mech. Anal., 231(1):45–61, (2019).
  89. K. Watanabe. Sur l’unicité du prolongement des solutions des équations elliptiques dégénérées. Tohoku Math. Journ. 34 (1982), 239-249.
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