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.

Pseudo Harmonic Measures and the Dirichlet Problem

1971 ◽  
Vol 23 (6) ◽  
pp. 1116-1120
Author(s):  
Maynard Arsove ◽  
Heinz Leutwiler
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Measure ◽  
Harmonic Functions ◽  
General Setting ◽  
Ideal Boundary ◽  
Jordan Curves ◽  
Boundary Properties ◽  
Harmonic Measures ◽  
Carry Over ◽  
Boundary Sets

For the case of plane regions bounded by finitely many disjoint Jordan curves, the solution of the Dirichlet problem can be expressed in terms of the classical harmonic measure of boundary arcs. At an appropriate stage in the development it is, in fact, useful to observe that the existence of such harmonic measures is equivalent to solvability of the Dirichlet problem (although one subsequently proves that all such regions are Dirichlet regions). We propose here to carry over this order of ideas to a quite general setting, in which arbitrary regions and ideal boundary structures are allowed. The counterparts of the classical harmonic measures of arcs are then harmonic functions with analogous boundary properties, but they no longer appear as measures in the boundary sets, in general. We shall refer to them as “pseudo harmonic measures”. Our main result shows how pseudo harmonic measures can be used to solve the Dirichlet problem.

scholarly journals Asymptotic tracts of harmonic functions II

1995 ◽  
Vol 38 (1) ◽  
pp. 35-52 ◽  
Author(s):  
K. F. Barth ◽  
D. A. Brannan
Keyword(s):  
Harmonic Function ◽  
Real Number ◽  
Real Function ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected ◽  
Prime End

An asymptotic tract of a real function u harmonic and non-constant in ℂ is a component of the set {z:u(z)≠c}, for some real number c; a quasi-tractT(≠ℂ) is an unbounded simply-connected domain in ℂ such that there exists a function u that is positive, unbounded and harmonic in T such that, for each point ζ∈∂T∩ℂ,and a ℱ-tract is an unbounded simply-connected domain T in ℂ whose every prime end that contains ∞ in its impression is of the first kind.The authors study the growth of a harmonic function in one of its asymptotic tracts, and the question of whether a quasi-tract is an asymptotic tract. The branching of either type of tract is also taken into consideration.

scholarly journals The Approximate Solution of 2D Dirichlet Problem in Doubly Connected Domains

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Atallah El-shenawy ◽  
Elena A. Shirokova
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Dirichlet Problem ◽  
Harmonic Function ◽  
Connected Domain ◽  
Boundary Value ◽  
Logarithmic Function ◽  
Cauchy Integral ◽  
Conjugate Harmonic Function ◽  
Doubly Connected Domain

We propose a new method for constructing an approximate solution of the two-dimensional Laplace equation in an arbitrary doubly connected domain with smooth boundaries for Dirichlet boundary conditions. Using the fact that the solution of the Dirichlet problem in a doubly connected domain is represented as the sum of a solution of the Schwarz problem and a logarithmic function, we reduce the solution of the Schwartz problem to the Fredholm integral equation with respect to the boundary value of the conjugate harmonic function. The solution of the integral equation in its turn is reduced to solving a linear system with respect to the Fourier coefficients of the truncated expansion of the boundary value of the conjugate harmonic function. The unknown coefficient of the logarithmic component of the solution of the Dirichlet problem is determined from the following fact. The Cauchy integral over the boundary of the domain with a density that is the boundary value of the analytical in this domain function vanishes at points outside the domain. The resulting solution of the Dirichlet problem is the sum of the real part of the Cauchy integral in the given domain and the logarithmic function. In order to avoid singularities of the Cauchy integral at points near the boundary, the solution at these points is replaced by a linear function. The resulting numerical solution is continuous in the domain up to the boundaries. Three examples of the solution of the Dirichlet problem are given: one example demonstrates the solution with constant boundary conditions in the domain with a complicated boundary; the other examples provide a comparison of the approximate solution with the known exact solution in a noncircular domain.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
M. T. Mustafa
Keyword(s):  
Harmonic Function ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Horizontal Component ◽  
Harmonic Morphisms

For Riemannian manifoldsMandN, admitting a submersionϕwith compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians onMandN, we determine conditions under which a harmonic function onU=ϕ−1(V)⊂Mprojects down, via its horizontal component, to a harmonic function onV⊂N.

scholarly journals Some remarks concerning harmonic functions on homogeneous graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Anders Karlsson
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Cayley Graphs ◽  
Group Cohomology ◽  
Radial Variation ◽  
International Audience

International audience We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.

A note on positive harmonic functions

1948 ◽  
Vol 44 (2) ◽  
pp. 289-291 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Duke Math ◽  
Negative Number ◽  
Image Position ◽  
American Math ◽  
Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

scholarly journals Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator

2019 ◽  
Vol 149 (6) ◽  
pp. 1577-1594
Author(s):  
Clifford Gilmore ◽  
Eero Saksman ◽  
Hans-Olav Tylli
Keyword(s):  
Growth Rate ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Optimal Growth ◽  
Differentiation Operator ◽  
Minimal Growth ◽  
Partial Differentiation

AbstractWe solve a problem posed by Blasco, Bonilla and Grosse-Erdmann in 2010 by constructing a harmonic function on ℝN, that is frequently hypercyclic with respect to the partial differentiation operator ∂/∂xk and which has a minimal growth rate in terms of the average L2-norm on spheres of radius r > 0 as r → ∞.

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