The hot spots problem in planar domains with on hole. (English) Zbl 1154.35330
Summary: There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
| [1] | R. Atar, Invariant wedges for a two-point reflecting Brownian motion and the “hot spots” problem , Electron. J. Probab. 6 (2001), no. 18. · Zbl 1125.60310 |
| [2] | R. Atar and K. Burdzy, On Neumann eigenfunctions in lip domains , J. Amer. Math. Soc. 17 (2004), 243–265. · Zbl 1151.35322 · doi:10.1090/S0894-0347-04-00453-9 |
| [3] | 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 |
| [4] | R. F. Bass, Probabilistic Techniques in Analysis , Probab. Appl. (N.Y.), Springer, New York, 1995. · Zbl 0817.60001 |
| [5] | R. F. Bass and K. Burdzy, Fiber Brownian motion and the “hot spots” problem , Duke Math. J. 105 (2000), 25–58. · Zbl 1006.60078 · doi:10.1215/S0012-7094-00-10512-1 |
| [6] | K. Burdzy and W. S. Kendall, Efficient Markovian couplings: Examples and counterexamples , Ann. Appl. Probab. 10 (2000), 362–409. · 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] | R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 , Interscience, New York, 1953. · Zbl 0053.02805 |
| [9] | G. B. Folland, Introduction to Partial Differential Equations , Math. Notes, Princeton Univ. Press, Princeton, 1976. · Zbl 0325.35001 |
| [10] | K. Itô and H. P. Mckean Jr., Diffusion Processes and Their Sample Paths , Grundlehren Math. Wiss. 125 , Springer, Berlin, 1974. |
| [11] | D. Jerison and N. Nadirashvili, The “hot spots” conjecture for domains with two axes of symmetry , J. Amer. Math. Soc. 13 (2000), 741–772. JSTOR: · Zbl 0948.35029 · doi:10.1090/S0894-0347-00-00346-5 |
| [12] | 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 |
| [13] | P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions , Comm. Pure Appl. Math. 37 (1984), 511–537. · Zbl 0598.60060 · doi:10.1002/cpa.3160370408 |
| [14] | M. N. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem , Trans. Amer. Math. Soc. 354 (2002), 4681–4702. JSTOR: · Zbl 1006.60077 · doi:10.1090/S0002-9947-02-03020-9 |
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.
