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The “hot spots” conjecture for a certain class of planar convex domains

J. Math. Phys. 50, 103530 (2009); doi:10.1063/1.3251335

Published 28 October 2009

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Yasuhito Miyamoto
Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
We prove the “hot spots” conjecture of Rauch [“Five problems: An introduction to the qualitative theory of partial differential equations,” Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics (Springer, Berlin, 1975), Vol. 446, pp. 355–369] for a certain class of planar convex domains. Specifically, we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the Neumann Laplacian on Omega attains its maximum (minimum) at points in [partial-derivative]Omega. One class of domains is the planar convex domain Omega satisfying diam(Omega)2/|Omega|<1.378. When Omega is a disk, diam(Omega)2/|Omega|[approximate]1.273. Hence, this condition indicates that Omega is a nearly circular planar convex domain. However, symmetries of the domain are not assumed. We give other sufficient conditions for domains for which the conjecture holds. We also give a new isoperimetric inequality. ©2009 American Institute of Physics
History: Received 28 August 2009; accepted 23 September 2009; published 28 October 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/103530/1
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0022-2488 (print)   1089-7658 (online)
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REFERENCES (21)
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  1. Atar, R. and Burdzy, K., “On Neumann eigenfunctions in lip domains,” J. Am. Math. Soc. 17, 243 (2004).
  2. Bañuelos, R. and Burdzy, K., “On the `hot spots' conjecture of J. Rauch,” J. Funct. Anal. 164, 1 (1999).
  3. Bauman, P., Carlson, N., and Phillips, D., “On the zeros of solutions to Ginzburg-Landau type systems,” SIAM J. Math. Anal. 24, 1283 (1993).
  4. Burdzy, K., “The hot spots problem in planar domains with one hole,” Duke Math. J. 129, 481 (2005).
  5. Burdzy, K. and Werner, W., “A counterexample to the `hot spots' conjecture,” Ann. Math. 149, 309 (1999).
  6. Carleman, T., “Sur les systèmes linéaires aux derivées partielles du premier ordre à deux variables,” Acad. Sci., Paris, C. R. 197, 471 (1933).
  7. Casten, R. and Holland, R., “Instability results for reaction diffusion equations with Neumann boundary conditions,” J. Differ. Equations 27, 266 (1978).
  8. Gurtin, M. E. and Matano, H., “On the structure of equilibrium phase transitions within the gradient theory of fluids,” Q. Appl. Math. 46, 301 (1988).
  9. Hartman, P. and Wintner, A., “On the local behavior of solutions of non-parabolic partial differential equations,” Am. J. Math. 75, 449 (1953).
  10. Helffer, B., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., and Owen, M. P., “Nodal sets for ground states of Schrödinger operators with zero magnetic field in non simply connected domains,” Commun. Math. Phys. 202, 629 (1999).
  11. Jerison, D. and Nadirashvili, N., “The `hot spots' conjecture for domains with two axes of symmetry,” J. Am. Math. Soc. 13, 741 (2000).
  12. Kawohl, B., Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics Vol. 1150 (Springer-Verlag, Berlin, 1985), p. 136.
  13. Matano, H., “Asymptotic behavior and stability of solutions of semilinear diffusion equations,” Publ. Res. Inst. Math. Sci. 15, 401 (1979).
  14. Miyamoto, Y., “An instability criterion for activator-inhibitor systems in a two-dimensional ball. II,” J. Differ. Equations 239, 61 (2007).
  15. Miyamoto, Y., “On the shape of stable patterns for activator-inhibitor systems in two dimensional domains,” Q. Appl. Math. 65, 357 (2007).
  16. Pascu, M., “Scaling coupling of reflecting Brownian motions and the hot spots problem,” Trans. Am. Math. Soc. 354, 4681 (2002).
  17. Payne, L. E., “On two conjectures in the fixed membrane eigenvalue problem,” Z. Angew. Math. Phys. 24, 721 (1973).
  18. Rauch, J., “Five problems: An introduction to the qualitative theory of partial differential equations,” Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics Vol. 446 (Springer, Berlin, 1975), pp. 355–369.
  19. Szegö, G., “Inequalities for certain eigenvalues of a membrane of given area,” J. Rational Mech. Anal. 3, 343 (1954).
  20. Weinberger, H. F., “An isoperimetric inequality for the n-dimensional free membrane problem,” J. Rational Mech. Anal. 5, 633 (1956).
  21. Yanagida, E. (private communication).

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