Location of hot spots in thin curved strips. (English) Zbl 1407.35069
The authors prove that the famous “hot-spot conjecture” for Neumann eigenvalues of the Laplacian holds for a certain class of domains, namely, thin strips about smooth curves in 2-dimensional oriented Riemannian manifolds.
The “hot-spot” conjecture can be stated as follows: \[ \forall x\in \Omega\,,\;\;\;\min_{\partial\Omega}u_2 < u_2(x)< \max_{\partial\Omega}u_2, \] where \(u_2\) is an eigenfunction associated with \(\lambda_2\), the first positive eigenvalue of the Neumann Laplacian on \(\Omega\), and \(\partial\Omega\) is the boundary of \(\Omega\).
Such conjecture holds for various classes of domains. However, there are counter-examples showing that the conjecture fails for certain domains of \(\mathbb R^n\) of manifolds.
The authors prove the validity of the conjecture for a quite large class of domains on 2-dimensional Riemannian manifolds. The domains considered in the present article are not covered by the classes for which already the conjecture is proved to hold.
The authors consider smooth curves \(\Gamma:[0,L]\rightarrow\mathcal A\), where \(\mathcal A\) is a smooth oriented 2-dimensional Riemannian manifold. The strip \(\Omega_{\varepsilon}\) about \(\Gamma\) of width \(2\varepsilon\) is the set of all points \(x\in\mathcal A\) for which there exists a geodesic of length less than \(\varepsilon\) from \(x\) which meets \(\Gamma\) orthogonally.
The authors prove that there exists a positive constant \(\varepsilon_0\) depending on \(\mathcal A\) and \(\Gamma\) such that for all \(\varepsilon\in]0,\varepsilon_0[\) the hot-spot conjecture holds for \(\Omega_{\varepsilon}\).
The result is a consequence of the fact that the authors are able to locate precisely all points of maximum and minimum of all Neumann eigenfunctions (not only \(u_2\)) of \(\Omega_{\varepsilon}\), for \(\varepsilon\) sufficiently small. This is achieved by comparing Neumann eigenfunctions and eigenvalues on the strip \(\Omega_{\varepsilon}\) with Neumann eigenfunctions and eigenvalues of the one-dimensional Neumann Laplacian on \((0,L)\), when \(\varepsilon\) is sufficiently small.
The “hot-spot” conjecture can be stated as follows: \[ \forall x\in \Omega\,,\;\;\;\min_{\partial\Omega}u_2 < u_2(x)< \max_{\partial\Omega}u_2, \] where \(u_2\) is an eigenfunction associated with \(\lambda_2\), the first positive eigenvalue of the Neumann Laplacian on \(\Omega\), and \(\partial\Omega\) is the boundary of \(\Omega\).
Such conjecture holds for various classes of domains. However, there are counter-examples showing that the conjecture fails for certain domains of \(\mathbb R^n\) of manifolds.
The authors prove the validity of the conjecture for a quite large class of domains on 2-dimensional Riemannian manifolds. The domains considered in the present article are not covered by the classes for which already the conjecture is proved to hold.
The authors consider smooth curves \(\Gamma:[0,L]\rightarrow\mathcal A\), where \(\mathcal A\) is a smooth oriented 2-dimensional Riemannian manifold. The strip \(\Omega_{\varepsilon}\) about \(\Gamma\) of width \(2\varepsilon\) is the set of all points \(x\in\mathcal A\) for which there exists a geodesic of length less than \(\varepsilon\) from \(x\) which meets \(\Gamma\) orthogonally.
The authors prove that there exists a positive constant \(\varepsilon_0\) depending on \(\mathcal A\) and \(\Gamma\) such that for all \(\varepsilon\in]0,\varepsilon_0[\) the hot-spot conjecture holds for \(\Omega_{\varepsilon}\).
The result is a consequence of the fact that the authors are able to locate precisely all points of maximum and minimum of all Neumann eigenfunctions (not only \(u_2\)) of \(\Omega_{\varepsilon}\), for \(\varepsilon\) sufficiently small. This is achieved by comparing Neumann eigenfunctions and eigenvalues on the strip \(\Omega_{\varepsilon}\) with Neumann eigenfunctions and eigenvalues of the one-dimensional Neumann Laplacian on \((0,L)\), when \(\varepsilon\) is sufficiently small.
Reviewer: Luigi Provenzano (Padova)
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 35B50 | Maximum principles in context of PDEs |
References:
| [1] | Atar, R.; Burdzy, K., On Neumann eigenfunctions in lip domains, J. Amer. Math. Soc., 17, 243-265 (2004) · Zbl 1151.35322 |
| [2] | Bañuelos, R.; Burdzy, K., On the “hot spots” conjecture of J. Rauch, J. Funct. Anal., 164, 1-33 (1999) · Zbl 0938.35045 |
| [3] | Bass, R. F.; Burdzy, K., Fiber Brownian motion and the “hot spots” problem, Duke Math. J., 105, 25-58 (2000) · Zbl 1006.60078 |
| [4] | Burdzy, K., The hot spots problem in planar domains with one hole, Duke Math. J., 129, 481-502 (2005) · Zbl 1154.35330 |
| [5] | Burdzy, K.; Werner, W., A counterexample to the “hot spots” conjecture, Ann. Math., 149, 309-317 (1999) · Zbl 0919.35094 |
| [6] | Freitas, P., Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces, Indiana Univ. Math. J., 51, 305-316 (2002) · Zbl 1026.58023 |
| [7] | Freitas, P.; Krejčiřík, D., Location of the nodal set for thin curved tubes, Indiana Univ. Math. J., 57, 1, 343-376 (2008) · Zbl 1187.35142 |
| [8] | Gray, A., Tubes (1990), Addison-Wesley Publishing Company: Addison-Wesley Publishing Company New York · Zbl 0692.53001 |
| [9] | Hartman, P., Geodesic parallel coordinates in the large, Amer. J. Math., 86, 705-727 (1964) · Zbl 0128.16105 |
| [10] | Jerison, D.; Nadirashvili, N., The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc., 13, 741-772 (2000) · Zbl 0948.35029 |
| [11] | Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag Berlin · Zbl 0148.12601 |
| [12] | Kawohl, B., Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, vol. 1150 (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0593.35002 |
| [13] | Krejčiřík, D., Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions, ESAIM Control Optim. Calc. Var., 15, 555-568 (2009) · Zbl 1173.35618 |
| [14] | Krejčiřík, D.; Tušek, M., Nodal sets of thin curved layers, J. Differential Equations, 258, 281-301 (2015) · Zbl 1323.35109 |
| [15] | Miyamoto, Y., The “hot spots” conjecture for a certain class of planar convex domains, J. Math. Phys., 50, Article 103530 pp. (2009) · Zbl 1283.35016 |
| [16] | Miyamoto, Y., A planar convex domain with many isolated “hot spots” on the boundary, Jpn. J. Ind. Appl. Math., 30, 145-164 (2013) · Zbl 1260.35090 |
| [17] | Pascu, M. N., Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc., 354, 4681-4702 (2002) · Zbl 1006.60077 |
| [18] | Post, O., Spectral Analysis on Graph-Like Spaces, Lecture Notes in Mathematics (2012), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1247.58001 |
| [19] | 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). Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Mathematics, vol. 446 (1975), Springer: Springer Berlin), 617-636 · Zbl 0312.35001 |
| [20] | Saitō, Y., Convergence of the Neumann Laplacian on shrinking domains, Analysis, 21, 171-204 (2001) · Zbl 0977.35039 |
| [21] | Schatzman, M., On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions, Appl. Anal., 61, 293-306 (1996) · Zbl 0865.35098 |
| [22] | Siudeja, B., Hot spots conjecture for a class of acute triangles, Math. Z., 280, 783-806 (2015) · Zbl 1335.35164 |
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