Superharmonicity of curvature function for the convex level sets of harmonic functions

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Xi-Nan Ma ◽  
Wei Zhang
Keyword(s):  
Level Sets ◽  
Harmonic Functions ◽  
Curvature Function

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 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

PRESCRIBING MEAN CURVATURE ON 𝔹n

2010 ◽  
Vol 21 (09) ◽  
pp. 1157-1187 ◽  
Author(s):  
WAEL ABDELHEDI ◽  
HICHEM CHTIOUI
Keyword(s):  
Critical Points ◽  
Mean Curvature ◽  
Level Sets ◽  
Curvature Function ◽  
Number Of Solutions ◽  
Total Contribution ◽  
Prescribed Mean Curvature ◽  
The Difference ◽  
Zero Scalar Curvature ◽  
Lagrange Functional

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹n, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

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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