Proof of the fundamental gap conjecture

Andrews, Ben; Clutterbuck, Julie

doi:10.1090/S0894-0347-2011-00699-1

Proof of the fundamental gap conjecture
HTML articles powered by AMS MathViewer

by Ben Andrews and Julie Clutterbuck

J. Amer. Math. Soc. 24 (2011), 899-916

DOI: https://doi.org/10.1090/S0894-0347-2011-00699-1

Published electronically: March 16, 2011

PDF | Request permission

Abstract:

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential. More generally, for an arbitrary smooth potential in higher dimensions, our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction, in terms of the diameter of the domain and a modulus of convexity for the potential.

References

Ben Andrews and Julie Clutterbuck, Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable, J. Differential Equations 246 (2009), no. 11, 4268–4283. MR 2517770, DOI 10.1016/j.jde.2009.01.024
Ben Andrews and Julie Clutterbuck, Time-interior gradient estimates for quasilinear parabolic equations, Indiana Univ. Math. J. 58 (2009), no. 1, 351–380. MR 2504416, DOI 10.1512/iumj.2009.58.3756
Mark S. Ashbaugh, The Fundamental Gap (2006), available at http://www.aimath.org/WWN/loweigenvalues/.
Mark S. Ashbaugh and Rafael Benguria, Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials, Proc. Amer. Math. Soc. 105 (1989), no. 2, 419–424. MR 942630, DOI 10.1090/S0002-9939-1989-0942630-X
Rodrigo Bañuelos and Pawel Kröger, Gradient estimates for the ground state Schrödinger eigenfunction and applications, Comm. Math. Phys. 224 (2001), no. 2, 545–550. MR 1869875, DOI 10.1007/s002200100551
Rodrigo Bañuelos and Pedro J. Méndez-Hernández, Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps, J. Funct. Anal. 176 (2000), no. 2, 368–399. MR 1784420, DOI 10.1006/jfan.2000.3611
M. van den Berg, On condensation in the free-boson gas and the spectrum of the Laplacian, J. Statist. Phys. 31 (1983), no. 3, 623–637. MR 711491, DOI 10.1007/BF01019501
Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
Burgess Davis, On the spectral gap for fixed membranes, Ark. Mat. 39 (2001), no. 1, 65–74. MR 1821082, DOI 10.1007/BF02388791
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
Evans M. Harrell, Double wells, Comm. Math. Phys. 75 (1980), no. 3, 239–261. MR 581948, DOI 10.1007/BF01212711
Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR 2251558, DOI 10.1007/3-7643-7706-2
Antoine Henrot and Michel Pierre, Variation et optimisation de formes, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 48, Springer, Berlin, 2005 (French). Une analyse géométrique. [A geometric analysis]. MR 2512810, DOI 10.1007/3-540-37689-5
Miklós Horváth, On the first two eigenvalues of Sturm-Liouville operators, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1215–1224. MR 1948113, DOI 10.1090/S0002-9939-02-06637-6
Nicholas J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983), no. 4, 603–614. MR 703287, DOI 10.1512/iumj.1983.32.32042
Pawel Kröger, On the spectral gap for compact manifolds, J. Differential Geom. 36 (1992), no. 2, 315–330. MR 1180385
Richard Lavine, The eigenvalue gap for one-dimensional convex potentials, Proc. Amer. Math. Soc. 121 (1994), no. 3, 815–821. MR 1185270, DOI 10.1090/S0002-9939-1994-1185270-4
Peter Li, A lower bound for the first eigenvalue of the Laplacian on a compact manifold, Indiana Univ. Math. J. 28 (1979), no. 6, 1013–1019. MR 551166, DOI 10.1512/iumj.1979.28.28075
Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR 573435
Jun Ling, Estimates on the lower bound of the first gap, Comm. Anal. Geom. 16 (2008), no. 3, 539–563. MR 2429968, DOI 10.4310/CAG.2008.v16.n3.a3
L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
I. M. Singer, Bun Wong, Shing-Tung Yau, and Stephen S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schrödinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 319–333. MR 829055
Robert G. Smits, Spectral gaps and rates to equilibrium for diffusions in convex domains, Michigan Math. J. 43 (1996), no. 1, 141–157. MR 1381604, DOI 10.1307/mmj/1029005394
S.-T. Yau, Gap of the first two eigenvalues of the Schrödinger operator with nonconvex potential, Mat. Contemp. 35 (2008), 267–285. MR 2584188
Shing-Tung Yau, Nonlinear analysis in geometry, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 33, L’Enseignement Mathématique, Geneva, 1986. Série des Conférences de l’Union Mathématique Internationale [Lecture Series of the International Mathematics Union], 8. MR 865650
Shing-Tung Yau, An estimate of the gap of the first two eigenvalues in the Schrödinger operator, Lectures on partial differential equations, New Stud. Adv. Math., vol. 2, Int. Press, Somerville, MA, 2003, pp. 223–235. MR 2055851
Qi Huang Yu and Jia Qing Zhong, Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator, Trans. Amer. Math. Soc. 294 (1986), no. 1, 341–349. MR 819952, DOI 10.1090/S0002-9947-1986-0819952-8
Jia Qing Zhong and Hong Cang Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27 (1984), no. 12, 1265–1273. MR 794292