Spectral minimal partitions of a sector

Virginie Bonnaillie-Noël; Corentin Léna

doi:10.3934/dcdsb.2014.19.27

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2014, Volume 19, Issue 1: 27-53. Doi: 10.3934/dcdsb.2014.19.27

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Spectral minimal partitions of a sector

Virginie Bonnaillie-Noël^1, and
Corentin Léna^2,

1.
IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av Robert Schuman, F-35170 Bruz
2.
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Bât. 425, F-91405 Orsay Cedex

Received: November 30, 2012

Revised: June 30, 2013

Published: December 2013

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Abstract

In this article, we are interested in determining spectral minimal $k$-partitions for angular sectors. We first deal with the nodal cases for which we can determine explicitly the minimal partitions. Then, in the case where the minimal partitions are not nodal domains of eigenfunctions of the Dirichlet Laplacian, we analyze the possible topologies of these minimal partitions. We first exhibit symmetric minimal partitions by using a mixed Dirichlet-Neumann Laplacian and then use a double covering approach to catch non symmetric candidates. In this way, we improve the known estimates of the energy associated with the minimal partitions.

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Mathematics Subject Classification: Primary: 35B05, 35J05, 49M25, 65F15, 65N25; Secondary: 65N30.

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