The size of the first eigenfunction
of a convex planar domain
Authors:
Daniel Grieser and David Jerison
Journal:
J. Amer. Math. Soc. 11 (1998), 41-72
MSC (1991):
Primary 35J25, 35B65; Secondary 35J05
DOI:
https://doi.org/10.1090/S0894-0347-98-00254-9
MathSciNet review:
1470858
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: This paper estimates the size of the first Dirichlet eigenfunction of a convex planar domain. The eigenfunction is shown to be well-approximated, uniformly for all convex domains, by the first Dirichlet eigenfunction of a naturally associated ordinary differential (Schrödinger) operator. In particular, the place where the eigenfunction attains its maximum is located to within a distance comparable to the inradius.
- [B1] Christer Borell, Hitting probabilities of killed Brownian motion: a study on geometric regularity, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 451–467. MR 777379
- [B2] Christer Borell, Greenian potentials and concavity, Math. Ann. 272 (1985), no. 1, 155–160. MR 794098, https://doi.org/10.1007/BF01455935
- [BL] 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, https://doi.org/10.1016/0022-1236(76)90004-5
- [CL] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Robert E. Krieger, Malabar, Florida, 1984, 1955 edition published by McGraw-Hill. MR 16:1022b
- [D] T. A. Driscoll, Eigenmodes of isospectral drums, SIAM Rev. 39 (1997), 1-17. CMP 97:09
- [GJ] Daniel Grieser and David Jerison, Asymptotics of the first nodal line of a convex domain, Invent. Math. 125 (1996), no. 2, 197–219. MR 1395718, https://doi.org/10.1007/s002220050073
- [J] David Jerison, The diameter of the first nodal line of a convex domain, Ann. of Math. (2) 141 (1995), no. 1, 1–33. MR 1314030, https://doi.org/10.2307/2118626
- [J1] D. Jerison, Locating the first nodal line in the Neumann problem, Trans. A. M. S. (to appear).
- [KP] Carlos E. Kenig and Jill Pipher, The ℎ-path distribution of the lifetime of conditioned Brownian motion for nonsmooth domains, Probab. Theory Related Fields 82 (1989), no. 4, 615–623. MR 1002903, https://doi.org/10.1007/BF00341286
- [K] Pawel Kröger, On the ground state eigenfunction of a convex domain in Euclidean space, Potential Anal. 5 (1996), no. 1, 103–108. MR 1373834, https://doi.org/10.1007/BF00276699
- [PW] Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825
- [PW1] M. H. Protter and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286-292.
- [S] 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, https://doi.org/10.1307/mmj/1029005394
Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 35J25, 35B65, 35J05
Retrieve articles in all journals with MSC (1991): 35J25, 35B65, 35J05
Additional Information
Daniel Grieser
Affiliation:
Humboldt Universität Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany
Email:
grieser@mathematik.hu-berlin.de
David Jerison
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
jerison@math.mit.edu
DOI:
https://doi.org/10.1090/S0894-0347-98-00254-9
Keywords:
Convex domains,
eigenfunctions
Received by editor(s):
February 17, 1997
Additional Notes:
The first author was a member of the Mathematical Sciences Research Institute, Berkeley. The second author was partially supported by NSF grants DMS-9401355 and DMS-9705825.
Article copyright:
© Copyright 1998
American Mathematical Society
