Isospectral Domains in Euclidean 3-Space

The question as to whether the shape of a drum can be heard has existed for around fifty years. The simple answer is ‘no’ as shown through the construction of isospectral domains. Isospectral domains are non-isometric domains that display the same spectra of frequencies of sound. These frequencies, deduced from the eigenvalues of the Laplacian, are determined by solving the wave equation in a domain omega , where alpha-omega is subject to Dirichlet boundary conditions. This paper presents methods to expand the already existing two dimensional transplantation proof into Euclidean 3-space and, through these means, provides a number of three dimensional isospectral domains.

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Explicit Solutions of the Wave Equation on Three Dimensional Space-Times: Two Examples with Dirichlet Boundary Conditions on a Disk

American Journal of Undergraduate Research ◽

10.33697/ajur.2006.005 ◽

2006 ◽

Vol 4 (4) ◽

Author(s):

Daniel Boykis ◽

Patrick Moylan

Keyword(s):

Wave Equation ◽

Euclidean Space ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Dimensional Space ◽

Three Dimensional ◽

Flat Space ◽

Dirichlet Boundary Conditions ◽

Dimensional Euclidean Space ◽

Curved Space

We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space.

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Charge algebra in Al(A)dSn spacetimes

Journal of High Energy Physics ◽

10.1007/jhep05(2021)210 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Adrien Fiorucci ◽

Romain Ruzziconi

Keyword(s):

Boundary Conditions ◽

Dirichlet Boundary ◽

Three Dimensional ◽

Dirichlet Boundary Conditions ◽

Boundary Structure ◽

Asymptotic Symmetry ◽

Renormalization Procedure ◽

Field Dependent ◽

Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

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