I-DREM: Relaxing the Square Integrability Condition

Abstract We continue the study of the one-loop partition function of AdS3 gravity with focus on the square-integrability condition on the fluctuating fields. In a previous work we found that the Brown-Henneaux boundary conditions follow directly from the L2 condition. Here we rederive the partition function as a ratio of Laplacian determinants by performing a suitable decomposition of the metric fluctuations. We pay special attention to the asymptotics of the fields appearing in the partition function. We also show that in the usual computation using ghost fields for the de Donder gauge, such gauge condition is accessible precisely for square-integrable ghost fields. Finally, we compute the spectrum of the relevant Laplacians in thermal AdS3, in particular noticing that there are no isolated eigenvalues, only essential spectrum. This last result supports the analytic continuation approach of David, Gaberdiel and Gopakumar. The purely essential spectra found are consistent with the independent results of Lee and Delay of the essential spectrum of the TT rank-2 tensor Lichnerowickz Laplacian on asymptotically hyperbolic spaces.

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On the Stability in Bonnet's Theorem of the Surface Theory

Georgian Mathematical Journal ◽

10.1515/gmj.2007.543 ◽

2007 ◽

Vol 14 (3) ◽

pp. 543-564

Author(s):

Yuri G. Reshetnyak

Keyword(s):

Differential Operators ◽

Sobolev Class ◽

Integrability Condition ◽

Complete Integrability ◽

Integral Representations ◽

Completely Integrable Systems ◽

Frobenius Theorem ◽

Fundamental Tensor ◽

The Stability ◽

Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

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Stability, Boundedness and Square Integrability Properties of Fourth Order Neutral Differential Equations with Variable Delay

2020 2nd International Conference on Mathematics and Information Technology (ICMIT) ◽

10.1109/icmit47780.2020.9047000 ◽

2020 ◽

Author(s):

Mebrouk RAHMANE ◽

Moussadek REMILI ◽

Larbi FATMI

Keyword(s):

Differential Equations ◽

Fourth Order ◽

Variable Delay ◽

Neutral Differential Equations ◽

Square Integrability

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An asymptotic treatment for non-convex fully nonlinear elliptic equations: Global Sobolev and BMO type estimates

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500530 ◽

2019 ◽

Vol 21 (07) ◽

pp. 1850053 ◽

Cited By ~ 1

Author(s):

J. V. da Silva ◽

G. C. Ricarte

Keyword(s):

Elliptic Equations ◽

A Priori Estimates ◽

Integrability Condition ◽

A Priori ◽

Nonlinear Elliptic Equations ◽

Borderline Case ◽

Fully Nonlinear Elliptic Equations ◽

Fully Nonlinear ◽

Text Type ◽

Nonlinear Elliptic

In this paper, we establish global Sobolev a priori estimates for [Formula: see text]-viscosity solutions of fully nonlinear elliptic equations as follows: [Formula: see text] by considering minimal integrability condition on the data, i.e. [Formula: see text] for [Formula: see text] and a regular domain [Formula: see text], and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting “fine” regularity estimates from a limiting operator, the Recession profile, associated to [Formula: see text] to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when [Formula: see text]. In such a scenery, we show that solutions admit [Formula: see text] type estimates for their second derivatives.

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