Estimates of the Green’s Function and its Regular Part on Heisenberg Group Domains

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Najoua Gamara ◽  
Habiba Guemri
Keyword(s):  
Heisenberg Group ◽  
Green’S Function ◽  
Green's Function ◽  
Forthcoming Paper ◽  
Regular Part ◽  
Kohn Laplacian ◽  
Preliminary Work ◽  
Characteristic Points ◽  
The Given ◽  
Euclidean Domains

AbstractThis paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s function for the Kohn Laplacian on domains far away from the set of characteristic points. We give some estimates of the Green’s function, its regular part and their derivatives analogous to those proved by A. Bahri, Y.Y. Li, O. Rey in [1], and O. Rey in [16] for Euclidean domains. While the study of such functions on the set of characteristic points of the given domain will be discussed in a forthcoming paper.

scholarly journals Green’s Function for a Slice of the Korányi Ball in the Heisenberg GroupHn

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Shivani Dubey ◽  
Ajay Kumar ◽  
Mukund Madhav Mishra
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group ◽  
Green’S Function ◽  
Green's Function ◽  
Representation Formula ◽  
Radial Symmetry ◽  
The Given ◽  
Explicit Expressions

We give a representation formula for solution of the inhomogeneous Dirichlet problem on the upper half Korányi ball and for the slice of the Korányi ball in the Heisenberg groupHnby obtaining explicit expressions of Green-like kernel when the given data has certain radial symmetry.

The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary

2004 ◽  
Vol 56 (3) ◽  
pp. 590-611
Author(s):  
Yilong Ni
Keyword(s):  
Riemannian Manifold ◽  
Heisenberg Group ◽  
Green’S Function ◽  
Heat Kernel ◽  
Green's Function ◽  
Beltrami Operator ◽  
Small Time ◽  
Time Estimates

AbstractWe study the Riemannian Laplace-Beltrami operator L on a Riemannian manifold with Heisenberg group H1 as boundary. We calculate the heat kernel and Green's function for L, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of H1. We also restrict L to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary.

scholarly journals Green’s function for certain domains in the Heisenberg group H n

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Mukund Madhav Mishra ◽  
Ajay Kumar ◽  
Shivani Dubey
Keyword(s):  
Heisenberg Group ◽  
Green’S Function ◽  
Green's Function

Asymptotics of the principal eigenvalue of the Laplacian in 2D periodic domains with small traps

2021 ◽  
pp. 1-28
Author(s):  
F. PAQUIN-LEFEBVRE ◽  
S. IYANIWURA ◽  
M.J WARD
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Principal Eigenvalue ◽  
Series Representation ◽  
Transcendental Equation ◽  
Iteration Scheme ◽  
Lattice Points ◽  
Regular Part ◽  
Asymptotic Result ◽  
Bravais Lattice

We derive and numerically implement various asymptotic approximations for the lowest or principal eigenvalue of the Laplacian with a periodic arrangement of localised traps of small \[\mathcal{O}(\varepsilon )\] spatial extent that are centred at the lattice points of an arbitrary Bravais lattice in \[{\mathbb{R}^2}\] . The expansion of this principal eigenvalue proceeds in powers of \[\nu \equiv - 1/\log (\varepsilon {d_c})\] , where d c is the logarithmic capacitance of the trap set. An explicit three-term approximation for this principal eigenvalue is derived using strong localised perturbation theory, with the coefficients in this series evaluated numerically by using an explicit formula for the source-neutral periodic Green’s function and its regular part. Moreover, a transcendental equation for an improved approximation to the principal eigenvalue, which effectively sums all the logarithmic terms in powers of v, is derived in terms of the regular part of the periodic Helmholtz Green’s function. By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved ‘log-summed’ approximation to the principal eigenvalue. From a numerical computation of the PDE eigenvalue problem defined on the fundamental Wigner–Seitz (WS) cell for the lattice, it is shown that the three-term asymptotic approximation for the principal eigenvalue agrees well with the numerical result only for a rather small trap radius. In contrast, the log-summed asymptotic result provides a very close approximation to the principal eigenvalue even when the trap radius is only moderately small. For a circular trap, the first few transcendental correction terms that further improves the log-summed approximation for the principal eigenvalue are derived. Finally, it is shown numerically that, amongst all Bravais lattices with a fixed area of the primitive cell, the principal eigenvalue is maximised for a regular hexagonal arrangement of traps.

GREEN'S FUNCTIONS ON FRACTALS

Fractals ◽  
2000 ◽  
Vol 08 (04) ◽  
pp. 385-402 ◽  
Author(s):  
JUN KIGAMI ◽  
DANIEL R. SHELDON ◽  
ROBERT S. STRICHARTZ
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problem ◽  
Computer Experiments ◽  
Recursive Formula ◽  
Absolute Maximum ◽  
Initial Value ◽  
Sharp Estimates ◽  
Self Similar ◽  
The Given

For a regular harmonic structure on a post-critically finite (p.c.f.) self-similar fractal, the Dirichlet problem for the Laplacian can be solved by integrating against an explicitly given Green's function. We give a recursive formula for computing the values of the Green's function near the diagonal, and use it to give sharp estimates for the decay of the Green's function near the boundary. We present data from computer experiments searching for the absolute maximum of the Green's function for two different examples, and we formulate two radically different conjectures for where the maximum occurs. We also investigate a local Green's function that can be used to solve an initial value problem for the Laplacian, giving an explicit formula for the case of the Sierpinski gasket. The local Green's function turns out to be unbounded, and in fact not even integrable, but because of cancelation, it is still possible to form a singular integral to solve the initial value problem if the given function satisfies a Hölder condition.

GREEN'S FUNCTION MATCHING METHOD FOR REALISTIC CALCULATIONS OF INTERFACES

1985 ◽  
Vol 46 (C4) ◽  
pp. C4-321-C4-329 ◽  
Author(s):  
E. Molinari ◽  
G. B. Bachelet ◽  
M. Altarelli
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Matching Method

THE ATOMISTIC GREEN'S FUNCTION METHOD FOR INTERFACIAL PHONON TRANSPORT

2014 ◽  
Vol 17 (N/A) ◽  
pp. 89-145 ◽  
Author(s):  
Sridhar Sadasivam ◽  
Yuhang Che ◽  
Zhen Huang ◽  
Liang Chen ◽  
Satish Kumar ◽  
...  
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Method ◽  
Phonon Transport ◽  
Green’S Function Method
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