scholarly journals Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

2009 ◽  
Vol 195 (3) ◽  
pp. 865-898 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Frédéric Robert
Keyword(s):  
Boundary Conditions ◽  
Green's Functions ◽  
Dirichlet Boundary ◽  
Green’S Functions ◽  
Dirichlet Boundary Conditions

scholarly journals Analytical Green’s functions for continuum spectra

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Eugenio Megías ◽  
Mariano Quirós
Keyword(s):  
Boundary Conditions ◽  
Green's Functions ◽  
Dirichlet Boundary ◽  
Green’S Functions ◽  
Neumann Boundary ◽  
Approximate Equation ◽  
New Physics ◽  
Dirichlet Boundary Conditions ◽  
Gauge Bosons ◽  
Linear Dilaton

Abstract Green’s functions with continuum spectra are a way of avoiding the strong bounds on new physics from the absence of new narrow resonances in experimental data. We model such a situation with a five-dimensional model with two branes along the extra dimension z, the ultraviolet (UV) and the infrared (IR) one, such that the metric between the UV and the IR brane is AdS5, thus solving the hierarchy problem, and beyond the IR brane the metric is that of a linear dilaton model, which extends to z → ∞. This simplified metric, which can be considered as an approximation of a more complicated (and smooth) one, leads to analytical Green’s functions (with a mass gap mg ∼ TeV and a continuum for s >$$ {m}_g^2 $$ m g 2 ) which could then be easily incorporated in the experimental codes. The theory contains Standard Model gauge bosons in the bulk with Neumann boundary conditions in the UV brane. To cope with electroweak observables the theory is also endowed with an extra custodial gauge symmetry in the bulk, with gauge bosons with Dirichlet boundary conditions in the UV brane, and without zero (massless) modes. All Green’s functions have analytical expressions and exhibit poles in the second Riemann sheet of the complex plane at s = $$ {M}_n^2 $$ M n 2 − iMnΓn, denoting a discrete (infinite) set of broad resonances with masses (Mn) and widths (Γn). For gauge bosons with Neumann or Dirichlet boundary conditions, the masses and widths of resonances satisfy the (approximate) equation s = −4$$ {m}_g^2{\mathcal{W}}_n^2 $$ m g 2 W n 2 [±(1 + i)/4], where $$ \mathcal{W} $$ W n is the n-th branch of the Lambert function.

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.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

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.

scholarly journals Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions

2021 ◽  
pp. 104123
Author(s):  
Firdous A. Shah ◽  
Mohd Irfan ◽  
Kottakkaran S. Nisar ◽  
R.T. Matoog ◽  
Emad E. Mahmoud
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Wavelet Method ◽  
Telegraph Equations
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