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A counterexample to the nodal domain conjecture and a related semilinear equation. (English) Zbl 0652.35085
The authors study the eigenvalue problem \[ -(\Delta +V)\phi +\lambda \phi =0\quad in\quad \Omega;\quad \phi =0\quad on\quad \partial \Omega, \] where V is a given smooth function, \(\lambda\) is an eigenvalue and \(\Omega\) is a bounded smooth domain in \(R^ n\), \(n\geq 2\). The set \(\{\) \(x\in \Omega |\) \(\phi (x)=0\}\) is called the nodal set of \(\phi\). The following conjecture has been around for quite some time.
Nodal Domain Conjecture: The nodal line of any second eigenfunction must intersect the boundary \(\partial \Omega\) at exactly two points. In the paper the authors construct a counterexample to show that the above conjecture is false when \(V\not\equiv 0\).
Reviewer: C.F.Wang

35P05 General topics in linear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
Full Text: DOI
[1] Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43 – 55. · Zbl 0334.35022
[2] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. · Zbl 0051.28802
[3] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209 – 243. · Zbl 0425.35020
[4] -S. Lin, On second eigenfunctions of the Laplacian in \( {{\mathbf{R}}^2}\), preprint.
[5] Wei-Ming Ni and Roger D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of \Delta \?+\?(\?,\?)=0, Comm. Pure Appl. Math. 38 (1985), no. 1, 67 – 108. · Zbl 0581.35021
[6] -M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations. The analomous case, Proc. Accad. Naz. Lincei 77 (1986), 231-257.
[7] Lawrence E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 721 – 729 (English, with German summary). · Zbl 0272.35058
[8] Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669 – 706. · Zbl 0479.53001
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