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

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**, 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****[CL]**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[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**, 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**, 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**, 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**, 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**, 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