On Nodal Lines of Neumann Eigenfunctions

Rami Atar (Technion - Israel Institute of Technology)
Krzysztof Burdzy (University of Washington)

Abstract


We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 129-139

Publication Date: June 3, 2002

DOI: 10.1214/ECP.v7-1055

References

  1. R. Atar, Invariant wedges for a two-point reflecting Brownian motion and the ``hot spots'' problem, Elect. J. of Probab. 6, (2001) paper 18, 1-19. Math Review article not available.
  2. R. Atar and K. Burdzy, On Neumann eigenfunctions in lip domains, (2002) preprint. Math Review article not available.
  3. C. Bandle, Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, 7. Pitman, Boston, Mass.-London, (1980). Math Review link
  4. R. Banuelos and K. Burdzy, On the ``hot spots'' conjecture of J. Rauch, J. Func. Anal. 164, (1999) 1-33. Math Review link
  5. R. Bass and K. Burdzy, Fiber Brownian motion and the `hot spots' problem, Duke Math. J. 105, (2000) 25-58. Math Review article not available.
  6. R. Bass, K. Burdzy and Z.-Q. Chen, Uniqueness for reflecting Brownian motion in lip domains, (2002) preprint. Math Review article not available.
  7. K. Burdzy and W. Kendall, Efficient Markovian couplings: examples and counterexamples, Ann. Appl. Probab. 10, (2000) 362-409. Math Review link
  8. K. Burdzy and W. Werner, A counterexample to the "hot spots" conjecture, Ann. Math. 149, (1999) 309-317. Math Review link
  9. D. Jerison, Locating the first nodal line in the Neumann problem. Trans. Amer. Math. Soc. 352, (2000) 2301-2317. Math Review link
  10. D. Jerison and N. Nadirashvili, The ``hot spots'' conjecture for domains with two axes of symmetry, J. Amer. Math. Soc. 13, (2000) 741-772. Math Review link
  11. B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics 1150, Springer, Berlin, (1985). Math Review link
  12. P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37, (1984), 511-537. Math Review link
  13. A. Melas, On the nodal line of the second eigenfunction of the Laplacian in R 2, J. Differential Geom. 35, (1992) 255-263. Math Review link
  14. N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, (1986) 281-282. Math Review link
  15. N.S. Nadirashvili, Multiple eigenvalues of the Laplace operator, Mathematics of the USSR, Sbornik, 133-134, (1988) 225-238. Math Review link
  16. M. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc. (2002) to appear. Math Review article not available.
  17. M. Pinsky, The eigenvalues of an equilateral triangle, SIAM J. Math. Anal. 11, (1980) 819-827. Math Review link
  18. M. Pinsky, Completeness of the eigenfunctions of the equilateral triangle, SIAM J. Math. Anal. 16, (1985) 848-851. Math Review link
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.