The Determinant of the Dirichlet-to-Neumann Map for Surfaces with Boundary

Abstract. We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli-Silvestre extension for (−∆) γ when γ ∈ (0, 1), and both a geometric interpretation and a curved analogue of the higher order extension found by R. Yang for (−∆) γ when γ > 1. We give three applications of this correspondence. First, we exhibit some energy identities for the fractional GJMS operators in terms of energies in the compactified Poincaré-Einstein manifold, including an interpretation as a renormalized energy. Second, for γ ∈ (1, 2), we show that if the scalar curvature and the fractional Q-curvature Q 2γ of the boundary are nonnegative, then the fractional GJMS operator P 2γ is nonnegative. Third, by assuming additionally that Q 2γ is not identically zero, we show that P 2γ satisfies a strong maximum principle.

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“…For this reason and due to its conformal covariance, it is thus natural to consider B to be the conformal half-Laplacian (cf. [23]). …”

Section: Introductionmentioning

confidence: 99%

On Fractional GJMS Operators

Case

Chang

2015

Comm Pure Appl Math

104

122

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“…It is hoped that using the vast amount of literature on the Dirichlet-to-Neumann map, (see, e.g., [43][44][45][46][47][48][49]), new insights can be gained on what the decisive topological and geometrical factors are that lead to attractive or repulsive Casimir forces.…”

Section: Discussionmentioning

confidence: 99%

Gluing Formula for Casimir Energies

Kirsten

Lee

2018

Symmetry

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Abstract:We provide a completely new perspective for the analysis of Casimir forces in very general piston configurations. To this end, in order to be self-contained, we prove a "gluing formula" well known in mathematics and relate it with Casimir forces in piston configurations. At the center of our description is the Dirichlet-to-Neumann operator, which encodes all the information about those forces. As an application, the results for previously considered piston configurations are reproduced in a streamlined fashion.

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“…Given any smooth compact manifold X with boundary and a metricḡ, it is possible to give a notion of the conformal Paneitz operator P γ with respect to the metric g and its associated curvature in this setting. We have (see also [Guillarmou and Guillopé 2007] for the case γ = 1 2 ): Proposition 4.1 [Chang and González 2011]. Let (X n+1 ,ḡ) be a compact smooth manifold with boundary.…”

Section: Weighted Normalized Volume In a General Settingmentioning

confidence: 99%