Open problems connected with level sets of harmonic functions

1988 ◽  
pp. 207-210 ◽  
Author(s):  
Bernhard Kawohl
Keyword(s):  
Level Sets ◽  
Harmonic Functions ◽  
Open Problems

scholarly journals Level sets of harmonic functions on the Sierpiński gasket

2002 ◽  
Vol 40 (2) ◽  
pp. 335-362 ◽  
Author(s):  
Anders Öberg ◽  
Robert S. Strichartz ◽  
Andrew Q. Yingst
Keyword(s):  
Level Sets ◽  
Harmonic Functions ◽  
Sierpinski Gasket ◽  
Sierpiński Gasket

A Topological Characterization of Conjugate Nets

1977 ◽  
Vol 29 (4) ◽  
pp. 707-721
Author(s):  
Paul A. Vincent
Keyword(s):  
Level Sets ◽  
Harmonic Functions ◽  
Function Theory ◽  
Topological Analysis ◽  
Topological Characterization ◽  
Families Of Curves ◽  
Open Maps

One aspect of topological analysis that authors, such as G. T. Whyburn and Marston Morse, have pointed to ([16; 6] for instance) as being fundamental in the development of function theory is the topological study of the level sets of analytic and harmonic functions or of their topological analogues, light open maps and pseudo-harmonic functions. The first step in this direction seems to have been made by H. Whitney [14] when he studied families of curves, given abstractly using a condition of regularity.

On p-harmonic functions, convex duality and an asymptotic formula for injection mould filling

1996 ◽  
Vol 7 (5) ◽  
pp. 417-437 ◽  
Author(s):  
Gunnar Aronsson
Keyword(s):  
Power Law ◽  
Level Sets ◽  
Harmonic Functions ◽  
Moving Boundary ◽  
Polymer Melt ◽  
Injection Point ◽  
Mould Filling ◽  
Power Law Exponent ◽  
Mould Cavity ◽  
Plastic Case

This work treats the injection of certain thermoplastics into a planar mould cavity. The problem is to determine the filling pattern. It is assumed that the thermoplastic can be modelled as a non-Newtonian fluid of power-law type whose power-law exponent is relatively small (the pseudo-plastic case). The dependence of the viscosity on thermal variations is neglected. The mathematical description leads to a moving boundary problem, for which an asymptotic solution is found. According to this solution, the expansion of the polymer melt follows the level sets of an interior distance function, which is determined by the geometry of the mould, and the position of the injection point. The solution is easily computed and results of numerical experiments are given.

scholarly journals Hausdorff dimension of level sets of generalized Takagi functions

2014 ◽  
Vol 157 (2) ◽  
pp. 253-278
Author(s):  
PIETER C. ALLAART
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Continuous Functions ◽  
Zero Set ◽  
Arbitrary Integer ◽  
Open Problems ◽  
Formula Group ◽  
Nowhere Differentiable Function ◽  
Image Position ◽  
Class Of Functions

AbstractThis paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form \begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*} where φ(x) is the distance from x to the nearest integer, and for each n, ωn is a {−1,1}-valued function which is constant on each interval [j/2n,(j+1)/2n), j=0,1,. . .,2n − 1. This class of functions includes Takagi's continuous but nowhere differentiable function. It is shown that the largest possible Hausdorff dimension of f−1(y) is $\log ((9+\sqrt{105})/2)/\log 16\approx .8166$, but in case each ωn is constant, the largest possible dimension is 1/2. These results are extended to the intersection of the graph of f with lines of arbitrary integer slope. Furthermore, two natural models of choosing the signs ωn(x) at random are considered, and almost-sure results are obtained for the Hausdorff dimension of the zero set and the set of maximum points of f. The paper ends with a list of open problems.

scholarly journals On convexity of level sets of p -harmonic functions

2013 ◽  
Vol 255 (7) ◽  
pp. 2065-2081 ◽  
Author(s):  
Ting Zhang ◽  
Wei Zhang
Keyword(s):  
Level Sets ◽  
Harmonic Functions

scholarly journals An almost rigidity theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth

2018 ◽  
Vol 22 (04) ◽  
pp. 1850076 ◽  
Author(s):  
Xian-Tao Huang
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Level Sets ◽  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Rigidity Theorem ◽  
Existence Problem ◽  
Maximal Volume ◽  
Almost Rigidity

The main results of this paper consist of two parts. First, we obtain an almost rigidity theorem which roughly says that on an [Formula: see text] space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Second, we apply this almost rigidity theorem to study noncompact [Formula: see text] spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence problem of nonconstant harmonic functions with polynomial growth on such [Formula: see text] spaces.

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