×

Fiber Brownian motion and the “hot spots” problem. (English) Zbl 1006.60078

This article is devoted to the proof of the following negative answer to the so-called “hot spots” conjecture: There exists a bounded Lipschitz domain in the plane such that its second Neumann eigenvalue is simple, and both extrema of the corresponding eigenfunction are attained only at interior points of the domain.
The domain taken as counterexample is made of two disjoint triangles whose vertices are connected by very thin tubes. The idea (coming from W. Werner) is that if a triangle is warm and the other is cold, then the centers of these triangles should remain respectively the warmest and coldest points of this domain. To proceed, a reflected Brownian motion is run in this domain, and is approximated by a “fiber Brownian motion”, corresponding to the domain modified by replacing the thin tubes by an uncountable collection of fibers. Such fiber process switches between 1- and 2-dimensional Brownian motions, and yields an approximation of the spectral theory of the domain considered. A lot of delicate technicity is necessry to achieve the arguments.

MSC:

60J65 Brownian motion
35P99 Spectral theory and eigenvalue problems for partial differential equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
60J45 Probabilistic potential theory
60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Bandle, Isoperimetric Inequalities and Applications , Monogr. Stud. Math. 7 , Pitman, Boston, 1980. · Zbl 0436.35063
[2] R. Bañuelos and K. Burdzy, On the “hot spots” conjecture of J. Rauch , J. Funct. Anal. 164 (1999), 1–33. · Zbl 0938.35045 · doi:10.1006/jfan.1999.3397
[3] R. F. Bass, Probabilistic Techniques in Analysis , Probab. Appl., Springer, New York, 1995. · Zbl 0817.60001
[4] R. F. Bass and M. T. Barlow, The construction of Brownian motion on the Sierpinski carpet , Ann. I. H. Poincaré Probab. Statist. 25 (1989), 225–257. · Zbl 0691.60070
[5] R. F. Bass and P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains , Ann. Probab. 19 (1991), 486–508. · Zbl 0732.60090 · doi:10.1214/aop/1176990437
[6] K. Burdzy and W. Kendall, Efficient Markovian couplings: Examples and counterexamples , to appear in Ann. Appl. Probab. · Zbl 0957.60083 · doi:10.1214/aoap/1019487348
[7] K. Burdzy and W. Werner, A counterexample to the “hot spots” conjecture , Ann. of Math. (2) 149 (1999), 309–317. JSTOR: · Zbl 0919.35094 · doi:10.2307/121027
[8] I. Chavel, Eigenvalues in Riemannian Geometry , Pure Appl. Math. 115 , Academic Press, Orlando, 1984. · Zbl 0551.53001
[9] K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths , Springer, Berlin, 1974. · Zbl 0837.60001
[10] D. Jerison and N. Nadirashvili, The “hot spots” conjecture for domains with two axes of symmetry , preprint, 1999. JSTOR: · Zbl 0948.35029 · doi:10.1090/S0894-0347-00-00346-5
[11] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE , Lecture Notes in Math. 1150 , Springer, Berlin, 1985. · Zbl 0593.35002 · doi:10.1007/BFb0075060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.