scholarly journals Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary

2021 ◽  
Author(s):  
Lehel Banjai ◽  
Lyonell Boulton
Keyword(s):  
Sharp Estimates ◽  
Planar Domains ◽  
Self Similar ◽  
Poincaré Constant

scholarly journals Sharp Estimates for Green’s Functions of Cone-Type Planar Domains

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Mohamed Amine Ben Boubaker ◽  
Mohamed Selmi
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Conformal Mappings ◽  
Sharp Estimates ◽  
Planar Domains ◽  
Cone Type

We establish sharp estimates for Green’s functions of cone-type planar domains. Our work generalizes all estimates given by Zhao in 1988 and Selmi in 2000. Our principal idea is to use conformal mappings.

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.

Self-similar collapse with non-radial motions

2006 ◽  
Vol 20 ◽  
pp. 1-4
Author(s):  
A. Nusser
Keyword(s):  
Self Similar
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