A natural Finsler-Laplace operator

Barthelmé, Thomas

doi:10.1007/s11856-012-0168-z

Cited by 21 publications

(19 citation statements)

References 30 publications

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“…It follows from the work by Kirchheim [22] and Ambrosio and Kirchheim [3] that rectifiable currents T with associated mass measures T are concentrated on a set that in a measure-theoretic sense is locally Finsler, and Lipschitz functions f on these sets are almost-everywhere tangentially differentiable, with tangential derivative d f . Even for Finsler spaces, there does not seem to be a canonical choice of a Laplace operator, see for instance the works by Bao and Lackey [5], Shen [29], Centore [10] and Barthelmé [7]. In Section 3, we will use the local Finsler structure and the tangential derivatives of Lipschitz functions to define a Dirichlet energy of Lipschitz functions f defined on an integral current T by…”

Section: Introductionmentioning

confidence: 99%

Semicontinuity of eigenvalues under intrinsic flat convergence

Portegies

2015

Calc. Var.

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ABSTRACT. We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. We define min-max values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the min-max values of the limit space are larger than or equal to the upper limit of the min-max values of the currents in the sequence. In particular, the infimum of the normalized energy is semicontinuous. On spaces that are infinitesimally Hilbertian, we can define a linear Laplace operator. We can show that semicontinuity under intrinsic flat convergence holds for eigenvalues below the essential spectrum, if the total volume of the spaces converges as well.

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Section: Introductionmentioning

confidence: 99%

Semicontinuity of eigenvalues under intrinsic flat convergence

Portegies

2015

Calc. Var.

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“…Recently, there are many mathematicians who have investigated properties of the eigenvalues of p-Laplacian on Finsler manifolds and Riemannian manifolds to estimate the spectrum in terms of the other geometric quantities of the manifold. (see [3,4,9,11,18,20]). …”

Section: Introductionmentioning

confidence: 99%

Eigenvalues Variation of the P-Laplacian Under the Ricci Flow on Sm

Azami¹

2017

JIMS

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Abstract. Let (M, F ) be a compact Finsler manifold. Studying the eigenvalues and eigenfunctions for the linear and nonlinear geometric operators is a known problem. In this paper we will consider the eigenvalue problem for the p-laplace operator for Sasakian metric acting on the space of functions on SM . We find the first variation formula for the eigenvalues of p-Laplacian on SM evolving by the Ricci flow on M and give some examples. Kami memperoleh rumus variasi pertama dan memberikan beberapa contoh untuk nilai eigen dari p-Laplacian pada SM yang melibatkan aliran Ricci pada M .Kata kunci: aliran Ricci, manifold Finsler, operator p-Laplace

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“…For instance, to our knowledge, the only known result about eigenvalues is given by Munteanu [16] in the case of Randers spaces. Following an idea of Patrick Foulon, the first author introduced in [3] another generalization of the Laplace operator which seems more approachable and that we study in this article.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we introduce the definitions and basic results that we will need. The main references for this section are [4,3].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue control for a Finsler–Laplace operator

Barthelmé

Colbois

2012

Ann Glob Anal Geom

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Using the definition of a Finsler-Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalue.

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A natural Finsler-Laplace operator

Cited by 21 publications

References 30 publications

Semicontinuity of eigenvalues under intrinsic flat convergence

Semicontinuity of eigenvalues under intrinsic flat convergence

Eigenvalues Variation of the P-Laplacian Under the Ricci Flow on Sm

Eigenvalue control for a Finsler–Laplace operator

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