Examples involving the geometric form of level curves of harmonic functions

1985 ◽  
Vol 7 (1) ◽  
pp. 480-485 ◽  
Author(s):  
A. Acker ◽  
L. Payne
Keyword(s):  
Harmonic Functions ◽  
Geometric Form ◽  
Level Curves

scholarly journals Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Tomasz Adamowicz ◽  
Giona Veronelli
Keyword(s):  
Harmonic Functions ◽  
Geodesic Curvature ◽  
Integral Curvature ◽  
Level Curve ◽  
Singular Surfaces ◽  
Constant Gaussian Curvature ◽  
Conical Singularities ◽  
Logarithmic Convexity ◽  
Level Curves ◽  
Laplace Type

AbstractWe investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives $$L'$$ L ′ and $$L''$$ L ′ ′ of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

scholarly journals On Level Curves of Harmonic and Analytic Functions on Riemann Surfaces

1969 ◽  
Vol 34 ◽  
pp. 77-87
Author(s):  
Shinji Yamashitad
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Vector Lattice ◽  
Level Curves ◽  
Harmonic Majorant ◽  
Harmonic And Analytic Functions ◽  
Hyperbolic Riemann Surface

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

Curvature of Level Curves of Harmonic Functions

1983 ◽  
Vol 26 (4) ◽  
pp. 399-405 ◽  
Author(s):  
Marvin Ortel ◽  
Walter Schneider
Keyword(s):  
Harmonic Function ◽  
Harmonic Measure ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Outer Boundary ◽  
Level Curves ◽  
Jordan Curves ◽  
Open Set ◽  
Doubly Connected Domain ◽  
Minimal Curvature

AbstractIf H is an arbitrary harmonic function defined on an open set Ω⊂ℂ, then the curvature of the level curves of H can be strictly maximal or strictly minimal at a point of Ω. However, if Ω is a doubly connected domain bounded by analytic convex Jordan curves, and if H is harmonic measure of Ω with respect to the outer boundary of Ω, then the minimal curvature of the level curves of H is attained on the boundary of Ω.

scholarly journals The level curves of harmonic functions

1966 ◽  
Vol 123 (2) ◽  
pp. 425-425 ◽  
Author(s):  
Leopold Flatto ◽  
Donald J. Newman ◽  
Harold S. Shapiro
Keyword(s):  
Harmonic Functions ◽  
Level Curves
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