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.

Sets of Polynomials Orthogonal Simultaneously on Four Ellipses

1968 ◽  
Vol 20 ◽  
pp. 1281-1294
Author(s):  
Ruth Goodman
Keyword(s):  
The Other ◽  
Level Curve ◽  
Entire Family ◽  
Level Curves ◽  
Concentric Circles

It has been shown by Walsh (3) and Szegö (2) that if a set of polynomials is orthogonal on both of two distinct curves, then one curve is a level curve of the other. Szegö (2) has determined all sets of polynomials which are orthogonal simultaneously on an entire family of level curves. There are five essentially different sets, two of which are orthogonal on concentric circles, and three of which are orthogonal on confocal ellipses. Merriman (1) has shown that the orthogonality of a set of polynomials on both of two concentric circles is sufficient to guarantee their orthogonality on the entire family of circles.

A new sea-level curve from Nova Scotia: evidence for a rapid acceleration of sea-level rise in the late mid-Holocene

1995 ◽  
Vol 32 (12) ◽  
pp. 2071-2080 ◽  
Author(s):  
D. B. Scott ◽  
K. Brown ◽  
E. S. Collins ◽  
F. S. Medioli
Keyword(s):  
South Carolina ◽  
Nova Scotia ◽  
Sea Level ◽  
Late Holocene ◽  
Atlantic Coast ◽  
Level Curve ◽  
Global Response ◽  
Level Curves ◽  
Ice Melt ◽  
Vertical Extent

A new late Holocene sea-level curve is presented from the Atlantic coast of Nova Scotia. Contrary to earlier data from the same area, this curve starts at 4400 sidereal years before present (BP) and shows a rapid acceleration between 4400 and 3800 BP, which coincides with a similar acceleration already reported from the Northumberland Strait (Nova Scotia) and an oscillation observed in South Carolina. Comparing the two Nova Scotia curves suggests that the acceleration lasts just over 1000 years and has a vertical extent of 10 m. One puzzling fact is that the 10 m vertical extent in Nova Scotia is 8 m more than the same event measured in South Carolina and it cannot be accounted for simply by postglacial isostatic depression, since that occurs on a much longer time scale. A closer examination of most of the sea-level curves from northeastern North America reveals that either the record is missing from this interval or it is inconsistent. We suggest that this acceleration is part of a global response that coincides with the end of the mid-Holocene warming period, possibly indicating a lag response between warming and ice melt.

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:

scholarly journals Area Properties of Strictly Convex Curves

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 391 ◽  
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Yoon-Tae Jung
Keyword(s):  
Partial Answer ◽  
Level Curve ◽  
Conic Sections ◽  
Curve Segment ◽  
Level Curves ◽  
Strictly Convex ◽  
Convex Curves ◽  
Constant Area ◽  
Characterization Theorems

We study functions defined in the plane E 2 in which level curves are strictly convex, and investigate area properties of regions cut off by chords on the level curves. In this paper we give a partial answer to the question: Which function has level curves whose tangent lines cut off from a level curve segment of constant area? In the results, we give some characterization theorems regarding conic sections.

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.

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