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.

Variation of Hausdorff dimension of Julia sets

1993 ◽  
Vol 13 (1) ◽  
pp. 167-174 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Hausdorff Dimension ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Analytic Family ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

AbstractLet (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfieswhere ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

Recovery of harmonic functions from partial boundary data respecting internal pointwise values

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Juliette Leblond ◽  
Dmitry Ponomarev
Keyword(s):  
Boundary Value Problem ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Boundary Value ◽  
Lipschitz Boundary ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Overdetermined Boundary Value Problem

Abstract We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary

A Variation of the Koebe Mapping in a Dense Subset of S

1987 ◽  
Vol 39 (1) ◽  
pp. 54-73 ◽  
Author(s):  
D. Bshouty ◽  
W. Hengartner
Keyword(s):  
Conformal Mapping ◽  
Uniform Convergence ◽  
Connected Domain ◽  
Dual Space ◽  
Dense Subset ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let H(U) be the linear space of holomorphic functions defined on the unit disk U endowed with the topology of normal (locally uniform) convergence. For a subset E ⊂ H(U) we denote by Ē the closure of E with respect to the above topology. The topological dual space of H(U) is denoted by H′(U).Let D, 0 ∊ D, be a simply connected domain in C. The unique univalent conformal mapping ϕ from U onto D, normalized by ϕ(0) = 0 and ϕ′(0) > 0 will be called “the Riemann Mapping onto D”. Let S be the set of all normalized univalent functions

scholarly journals ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

2017 ◽  
Vol 18 (3) ◽  
pp. 591-618 ◽  
Author(s):  
Juhan Aru ◽  
Avelio Sepúlveda ◽  
Wendelin Werner
Keyword(s):  
Harmonic Function ◽  
Connected Domain ◽  
Free Field ◽  
Mean Value ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Simply Connected Domain ◽  
The Mean ◽  
Conformal Loop Ensemble ◽  
Simply Connected

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.

On positive harmonic functions

1952 ◽  
Vol 48 (4) ◽  
pp. 571-577
Author(s):  
A. C. Allen
Keyword(s):  
Real Axis ◽  
Half Plane ◽  
Harmonic Functions ◽  
Regular Function ◽  
Open Interval ◽  
Negative Number ◽  
The Real ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Any harmonic function which is defined and positive in the half-plane η > 0 may be expressed bywhere C is a non-negative number, and G(x) is a bounded non-decreasing function. For a simple proof see Loomis and Widder (2). Let us writewhere w(z) is a regular function of z in η > 0, and satisfies the following conditions: (i) w(z) is real and continuous at all points of the open interval (a, b) of the real axis [the interval may be unbounded]; (ii) there exists a simply connected domain Δ, lying in the half-plane η > 0, whose frontier contains the interval (a, b) of the real axis and which is mapped ‘simply’ on the half-plane υ > 0 by the conformal transformation w = w(z).

scholarly journals Asymptotic Solution Of Differential Equations In a Domain Containing a Regular Singular Point

1956 ◽  
Vol 8 ◽  
pp. 97-104 ◽  
Author(s):  
N. D. Kazarinoff ◽  
R. McKelvey
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Singular Point ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Connected Domain ◽  
Regular Singular Point ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

1. Introduction. In this paper we study the asymptotic behavior in λ of the solutions about the origin in the z-plane of the differential equation.Both the variable z and the parameter λ are complex. The coefficient P(z, λ) is assumed to be analytic and single-valued in λ at infinity and in z throughout a bounded, closed, simply connected domain D containing z = 0.

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).

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