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

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.

scholarly journals Isospectral Domains in Euclidean 3-Space

2012 ◽  
Vol 11 (1 and 2) ◽  
Author(s):  
Christopher Cox
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Alpha Omega ◽  
Eigenvalues Of The Laplacian

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.

scholarly journals Charge algebra in Al(A)dSn spacetimes

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.

LP -bound for the Fourier Transform of Surface-Carried Measures Supported on Hypersurfaces with D∞ Type Singularities

2020 ◽  
pp. 350-359
Author(s):  
Nigina A. Soleeva
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Convex Surfaces ◽  
The Fourier Transform

Estimate for Fourier transform of surface-carried measures supported on non-convex surfaces of three-dimensional Euclidean space is considered in this paper.The exact convergence exponent was found wherein the Fourier transform of measures is integrable in tree-dimensional space. This result gives an answer to the question posed by Erd¨osh and Salmhofer

scholarly journals JACOBI ELLIPTIC SOLUTIONS OF λϕ4 THEORY IN A FINITE DOMAIN

2000 ◽  
Vol 15 (17) ◽  
pp. 2645-2659 ◽  
Author(s):  
J. A. ESPICHÁN CARRILLO ◽  
A. MAIA ◽  
V. M. MOSTEPANENKO
Keyword(s):  
Energy Density ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dimensional Space ◽  
Elliptic Functions ◽  
Dirichlet Boundary Conditions ◽  
Finite Domain ◽  
Dimensional Space Time ◽  
Elliptic Solutions ◽  
Minimal Mass

The general static solutions of the scalar field equation for the potential [Formula: see text] are determined for a finite domain in (1+1)-dimensional space–time. A family of real solutions is described in terms of Jacobi Elliptic Functions. We show that the vacuum–vacuum boundary conditions can be reached by elliptic cn-type solutions in a finite domain, such as that of the Kink, for which they are imposed at infinity. We prove uniqueness for elliptic sn-type solutions satisfying Dirichlet boundary conditions in a finite interval (box) as well the existence of a minimal mass corresponding to these solutions in a box. We defined expressions for the "topological charge," "total energy" (or classical mass) and "energy–density" for elliptic sn-type solutions in a finite domain. For large length of the box the conserved charge, classical mass and energy density of the Kink are recovered. Also, we have shown that using periodic boundary conditions the results are the same as in the case of Dirichlet boundary conditions. In the case of antiperiodic boundary conditions all elliptic sn-type solutions are allowed.

On seismic deghosting using integral representation for the wave equation: Use of Green’s functions with Neumann or Dirichlet boundary conditions

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. T89-T98 ◽  
Author(s):  
Lasse Amundsen ◽  
Arthur B. Weglein ◽  
Arne Reitan
Keyword(s):  
Wave Equation ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Representation Theorem ◽  
Green's Functions ◽  
Dirichlet Boundary ◽  
Green’S Functions ◽  
Dirichlet Boundary Conditions ◽  
Sea Surface ◽  
Receiver Side

The recent interest in broadband seismic technology has spurred research into new and improved seismic deghosting solutions. One starting point for deriving deghosting methods is the representation theorem, which is an integral representation for the wave equation. Recent research results show that by using Green’s functions with Dirichlet boundary conditions in the representation theorem, source-side deghosting of already receiver-side deghosted wavefields can be achieved. We found that the choice of Green’s functions with Neumann boundary conditions on the sea surface and the plane that contains the sources leads to an identical but simpler solution with fewer processing steps. In addition, we found that pressure data can be receiver-side deghosted by introducing Green’s functions with Dirichlet boundary conditions on the sea surface and the plane containing the receivers into a modified representation theorem. The deghosting methods derived from the representation theorem are wave-theoretic algorithms defined in the frequency-space domain and can accommodate streamers of any shape (e.g., slanted). Our theoretical analysis of deghosting is performed in the frequency-wavenumber domain where analytical deghosting solutions are well known and thus are available for verifying the solutions. A simple numerical example can be used to show how source-side deghosting can be performed in the space domain by convolving data with Green’s functions.

Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra

2020 ◽  
Author(s):  
Vitoriano Ruas
Keyword(s):  
Boundary Conditions ◽  
Error Estimates ◽  
Degrees Of Freedom ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Dirichlet Problems ◽  
Simple Alternative ◽  
Curved Domains

Abstract In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

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