Consider two domains connected by a thin tube: it can be shown that, generically, the mass of a given eigenfunction of the Dirichlet Laplacian concentrates in only one of them. The restriction to the other domain, when suitably normalized, develops a singularity at the junction of the tube, as the channel section tends to zero. Our main result states that, under a nondegeneracy condition, the normalized limiting profile has a singularity of order N-1, where N is the space dimension. We give a precise description of the asymptotic behavior of eigenfunctions at the singular junction, which provides us with some important information about its sign near the tunnel entrance. More precisely, the solution is shown to be one-sign in a neighborhood of the singular junction. In other words, we prove that the nodal set does not enter inside the channel

Felli, V., Terracini, S. (2013). Singularity of eigenfunctions at the junction of shrinking tubes, Part I. JOURNAL OF DIFFERENTIAL EQUATIONS, 255(4), 633-700 [10.1016/j.jde.2013.04.017].

Singularity of eigenfunctions at the junction of shrinking tubes, Part I

FELLI, VERONICA;
2013

Abstract

Consider two domains connected by a thin tube: it can be shown that, generically, the mass of a given eigenfunction of the Dirichlet Laplacian concentrates in only one of them. The restriction to the other domain, when suitably normalized, develops a singularity at the junction of the tube, as the channel section tends to zero. Our main result states that, under a nondegeneracy condition, the normalized limiting profile has a singularity of order N-1, where N is the space dimension. We give a precise description of the asymptotic behavior of eigenfunctions at the singular junction, which provides us with some important information about its sign near the tunnel entrance. More precisely, the solution is shown to be one-sign in a neighborhood of the singular junction. In other words, we prove that the nodal set does not enter inside the channel
Articolo in rivista - Articolo scientifico
Weighted elliptic eigenvalue problem, dumbbell domains, Almgren monotonicity formula.
English
2013
255
4
633
700
open
Felli, V., Terracini, S. (2013). Singularity of eigenfunctions at the junction of shrinking tubes, Part I. JOURNAL OF DIFFERENTIAL EQUATIONS, 255(4), 633-700 [10.1016/j.jde.2013.04.017].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/44671
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