Eigenvalues of the Laplacian on domains with fractal boundary

2019 ◽  
pp. 267-277
Author(s):  
Paul Pollack ◽  
Carl Pomerance
Keyword(s):  
Fractal Boundary ◽  
Eigenvalues Of The Laplacian

A biregular extension theorem from a fractal surface in

2011 ◽  
Vol 18 (1) ◽  
pp. 21-29
Author(s):  
Ricardo Abreu Blaya ◽  
Juan Bory Reyes ◽  
Tania Moreno García
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Extension Theorem ◽  
Fractal Surface ◽  
Fractal Boundary ◽  
Hölder Continuous

Abstract The aim of this paper is to prove the characterization on a bounded domain of with fractal boundary and a Hölder continuous function on the boundary guaranteeing the biregular extendability of the later function throughout the domain.

Fractal Boundary Explorations for a Nonlinear Two Degree-of-Freedom System

2012 ◽  
Author(s):  
Benjamin A. M. Owens ◽  
Brian P. Mann
Keyword(s):  
Initial Conditions ◽  
Coupled System ◽  
Basins Of Attraction ◽  
Degree Of Freedom ◽  
Fractal Boundary ◽  
Final State ◽  
Potential Wells ◽  
Mechanical Coupling ◽  
Mass Spring ◽  
Two Degree Of Freedom

This paper explores a two degree-of-freedom nonlinearly coupled system with two distinct potential wells. The system consists of a pair of linear mass-spring-dampers with a non-linear, mechanical coupling between them. This nonlinearity creates fractal boundaries for basins of attraction and forced well-escape response. The inherent uncertainty of these fractal boundaries is quantified for errors in the initial conditions and parameter space. This uncertainty relationship provides a measure of the final state and transient sensitivity of the system.

scholarly journals On purely periodic beta-expansions of Pisot numbers

2002 ◽  
Vol 166 ◽  
pp. 183-207 ◽  
Author(s):  
Yuki Sano
Keyword(s):  
Natural Extension ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Pisot Number ◽  
Fractal Boundary ◽  
Pisot Numbers ◽  
Dimensional Domain

AbstractWe characterize numbers having purely periodic β-expansions where β is a Pisot number satisfying a certain irreducible polynomial. The main tool of the proof is to construct a natural extension on a d-dimensional domain with a fractal boundary.

scholarly journals Diffusion and propagation problems in some ramified domains with a fractal boundary

2006 ◽  
Vol 40 (4) ◽  
pp. 623-652 ◽  
Author(s):  
Yves Achdou ◽  
Christophe Sabot ◽  
Nicoletta Tchou
Keyword(s):  
Fractal Boundary

scholarly journals The Effect of Indentation Angle of Koch Fractal Boundary on the Performance of Microstrip Antenna

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
P. Nageswara Rao ◽  
N. V. S. N. Sarma
Keyword(s):  
Resonant Frequency ◽  
Experimental Verification ◽  
Microstrip Antenna ◽  
Fractal Boundary ◽  
Triangular Patch ◽  
Resonant Behavior ◽  
Koch Fractal

The effect of indentation angle of Koch fractal boundary applied to a triangular patch on the resonant behavior and bandwidth is presented. It is shown that the resonant frequency can be controlled by changing the indentation angle of the boundary. With the experimental verification, it is established that for an indentation angle of more bandwidth is obtained compared to conventional .

scholarly journals Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk

2010 ◽  
Vol 62 (4) ◽  
pp. 808-826
Author(s):  
Eveline Legendre
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
First Eigenvalue ◽  
Bounded Number ◽  
Eigenvalues Of The Laplacian ◽  
Neumann Laplacian ◽  
Dirichlet And Neumann Conditions ◽  
Second Eigenvalue

AbstractWe study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

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