On Neumann eigenfunctions in lip domains

Atar, Rami; Burdzy, Krzysztof

doi:10.1090/S0894-0347-04-00453-9

On Neumann eigenfunctions in lip domains
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by Rami Atar and Krzysztof Burdzy;

J. Amer. Math. Soc. 17 (2004), 243-265

DOI: https://doi.org/10.1090/S0894-0347-04-00453-9

Published electronically: February 11, 2004

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Abstract:

A “lip domain” is a planar set lying between graphs of two Lipschitz functions with constant 1. We show that the second Neumann eigenvalue is simple in every lip domain except the square. The corresponding eigenfunction attains its maximum and minimum at the boundary points at the extreme left and right. This settles the “hot spots” conjecture for lip domains as well as two conjectures of Jerison and Nadirashvili. Our techniques are probabilistic in nature and may have independent interest.

References

Omer Adelman, Krzysztof Burdzy, and Robin Pemantle, Sets avoided by Brownian motion, Ann. Probab. 26 (1998), no. 2, 429–464. MR 1626170, DOI 10.1214/aop/1022855639
Rami Atar, Invariant wedges for a two-point reflecting Brownian motion and the “hot spots” problem, Electron. J. Probab. 6 (2001), no. 18, 19. MR 1873295, DOI 10.1214/EJP.v6-91
Rodrigo Bañuelos and Krzysztof Burdzy, On the “hot spots” conjecture of J. Rauch, J. Funct. Anal. 164 (1999), no. 1, 1–33. MR 1694534, DOI 10.1006/jfan.1999.3397
Richard F. Bass and Krzysztof Burdzy, Lifetimes of conditioned diffusions, Probab. Theory Related Fields 91 (1992), no. 3-4, 405–443. MR 1151804, DOI 10.1007/BF01192065
Richard F. Bass and Krzysztof Burdzy, Fiber Brownian motion and the “hot spots” problem, Duke Math. J. 105 (2000), no. 1, 25–58. MR 1788041, DOI 10.1215/S0012-7094-00-10512-1
Richard F. Bass and Pei Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991), no. 2, 486–508. MR 1106272
Krzysztof Burdzy and Zen-Qing Chen, Weak convergence of reflecting Brownian motions, Electron. Comm. Probab. 3 (1998), 29–33. MR 1625707, DOI 10.1214/ECP.v3-990
Krzysztof Burdzy and Zhen-Qing Chen, Coalescence of synchronous couplings, Probab. Theory Related Fields 123 (2002), no. 4, 553–578. MR 1921013, DOI 10.1007/s004400200202
Krzysztof Burdzy and Wilfrid S. Kendall, Efficient Markovian couplings: examples and counterexamples, Ann. Appl. Probab. 10 (2000), no. 2, 362–409. MR 1768241, DOI 10.1214/aoap/1019487348
Krzysztof Burdzy and Wendelin Werner, A counterexample to the “hot spots” conjecture, Ann. of Math. (2) 149 (1999), no. 1, 309–317. MR 1680567, DOI 10.2307/121027
Kai Lai Chung, The lifetime of conditional Brownian motion in the plane, Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), no. 4, 349–351 (English, with French summary). MR 771894
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
Rainer Hempel, Luis A. Seco, and Barry Simon, The essential spectrum of Neumann Laplacians on some bounded singular domains, J. Funct. Anal. 102 (1991), no. 2, 448–483. MR 1140635, DOI 10.1016/0022-1236(91)90130-W
David Jerison and Nikolai Nadirashvili, The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc. 13 (2000), no. 4, 741–772. MR 1775736, DOI 10.1090/S0894-0347-00-00346-5
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619, DOI 10.1007/BFb0075060
Wilfrid S. Kendall, Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel, J. Funct. Anal. 86 (1989), no. 2, 226–236. MR 1021137, DOI 10.1016/0022-1236(89)90053-0
Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR 958914
P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984), no. 4, 511–537. MR 745330, DOI 10.1002/cpa.3160370408
Mihai N. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4681–4702. MR 1926894, DOI 10.1090/S0002-9947-02-03020-9
Philip Protter, Stochastic integration and differential equations, Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 1990. A new approach. MR 1037262, DOI 10.1007/978-3-662-02619-9
Feng Yu Wang, Application of coupling methods to the Neumann eigenvalue problem, Probab. Theory Related Fields 98 (1994), no. 3, 299–306. MR 1262968, DOI 10.1007/BF01192256