The hot spots conjecture can be false: some numerical examples. (English) Zbl 1487.35204
The author gives a procedure to construct domains with one hole for which the hot spots conjecture can be shown numerically to be false. The hot spots conjecture stems from J. A. Goldstein (ed.) [Partial differential equations and related topics. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0293.00013)] and was formulated later explicitly by B. Kawohl [Rearrangements and convexity of level sets in PDE. Berlin etc.: Springer-Verlag (1985; Zbl 0593.35002)]. It states that, for an open, connected and bounded domain \(D\subset\mathbb{R}^2\) with Lipschitz boundary \(\Gamma\), the eigenfunction(s) corresponding to the first non-trivial Neumann eigenvalue of the 2-dimensional Laplacian on \(D\) achieve(s) a global minimum and maximum on the boundary \(\Gamma\).
The conjecture has been shown to hold for several classes of convex domains with symmetries and it is believed to hold true for arbitrary convex domains. However, some results of [K. Burdzy and W. Werner, Ann. Math., 149, 309–317 (1999; Zbl 0919.35094); R.F. Bass and K. Burdzy, Duke Math. J. 105, No. 1, 25–58 (2000; Zbl 1006.60078); K. Burdzy, Duke Math. J. 129, No. 3, 481–502 (2005; Zbl 1154.35330)] have shown that there are domains with one or more holes for which the conjecture is false. The proof is technical and based on stochastic arguments and the purpose of the current paper is to construct simpler domains and show numerically that the conjecture is false for such domains. The method presented is this paper is based on a reformulation of the eigenvalue problem in terms of a boundary integral equation and a numerical approximation using boundary element collocation method. From this, non-trivial Neumann eigenvalues and their corresponding eigenfunctions are computed with high precision using contour integrals of the resolvent operator.
After presenting several domains for which the hot spots conjecture holds, the author constructs domains with one hole and thin ‘handles’ for which the hot spots conjecture is false. In view of the various examples, the author conjectures that exact symmetry is not necessary for the failure of the hot spots conjecture, but that domains which are far from being symmetric should satisfy the conjecture.
In complement to the paper, the author provides the Matlab codes used in the computations.
The conjecture has been shown to hold for several classes of convex domains with symmetries and it is believed to hold true for arbitrary convex domains. However, some results of [K. Burdzy and W. Werner, Ann. Math., 149, 309–317 (1999; Zbl 0919.35094); R.F. Bass and K. Burdzy, Duke Math. J. 105, No. 1, 25–58 (2000; Zbl 1006.60078); K. Burdzy, Duke Math. J. 129, No. 3, 481–502 (2005; Zbl 1154.35330)] have shown that there are domains with one or more holes for which the conjecture is false. The proof is technical and based on stochastic arguments and the purpose of the current paper is to construct simpler domains and show numerically that the conjecture is false for such domains. The method presented is this paper is based on a reformulation of the eigenvalue problem in terms of a boundary integral equation and a numerical approximation using boundary element collocation method. From this, non-trivial Neumann eigenvalues and their corresponding eigenfunctions are computed with high precision using contour integrals of the resolvent operator.
After presenting several domains for which the hot spots conjecture holds, the author constructs domains with one hole and thin ‘handles’ for which the hot spots conjecture is false. In view of the various examples, the author conjectures that exact symmetry is not necessary for the failure of the hot spots conjecture, but that domains which are far from being symmetric should satisfy the conjecture.
In complement to the paper, the author provides the Matlab codes used in the computations.
Reviewer: Gilles Evéquoz (Delémont)
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 65M38 | Boundary element methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
interior Neumann eigenvalues; Helmholtz equation; potential theory; boundary integral equations; numericsReferences:
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