Principal eigenvalue of a very badly degenerate operator and applications

Juutinen, Petri

doi:10.1016/j.jde.2007.01.020

Cited by 57 publications

(61 citation statements)

References 20 publications

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“…Concerning the limit as p → ∞ for the eigenvalue problem for the p−Laplacian in the usual PDE case we refer to [3,4,13,14,27]. To study this limit the main point is to use adequate test functions to obtain bounds that are uniform in p in order to gain compactness on a sequence of eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

Clustering for metric graphs using the p-Laplacian

Pezzo¹,

Rossi²

2016

Michigan Math. J.

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Abstract. We deal with the clustering problem in a metric graph. We look for two clusters, to this end we study the first non-zero eigenvalue of the p−Laplacian on a quantum graph with Newmann or Kirchoff boundary conditions on the nodes. Then, an associated eigenfunction up provides two sets inside the graph, {up > 0} and {up < 0} that define the clusters. Moreover, we describe in detail the limit cases p → ∞ and p → 1.

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Section: Introductionmentioning

confidence: 99%

Clustering for metric graphs using the p-Laplacian

Pezzo¹,

Rossi²

2016

Michigan Math. J.

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“…It was also discovered by Berestycki [1] for Sturm-Liouville equations. For more on principal eigenvalues of nonlinear elliptic operators, we refer to [6,7,19,27,29,30].…”

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confidence: 99%

Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

Armstrong

2009

Journal of Differential Equations

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We study the fully nonlinear elliptic equationin a smooth bounded domain Ω, under the assumption that the nonlinearity F is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal "half" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the "second" eigenvalue is positive, and generalize the anti-maximum principle of Clément and Peletier [P. Clément, L.A. Peletier, An antimaximum principle for second-order elliptic operators, J. Differential Equations 34 (2) (1979) 218-229] to homogeneous, fully nonlinear operators.

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“…The concept of principal eigenvalue for boundary value problems of elliptic operators has been extended, in the last decades, to quasi-nonlinear and fully-nonlinear equations (somehow "abusing" the name of eigenvalue) see [1,2,26,28,29,22,7,23], etc. In all the cases we know, two features of the operators are requested, homogeneity and ellipticity.…”

Section: Introductionmentioning

confidence: 99%