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Spectral minimal partitions of a sector. (English) Zbl 1286.35175

Summary: 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.

MSC:

35P05 General topics in linear spectral theory for PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

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