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

scholarly journals Computing Robin Problem on Unbounded Simply Connected Domain via an Integral Equation with the Generalized Neumann Kernel

2017 ◽  
Vol 2 (3) ◽  
pp. 120-124
Author(s):  
Shwan H. H. Al-Shatri ◽  
Karzan Wakil ◽  
Munira Ismail
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Connected Domain ◽  
Boundary Value ◽  
Solution Technique ◽  
Robin Problem ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Robin Boundary ◽  
Robin Boundary Value Problem

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation, and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth.

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 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 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.

Sign in / Sign up

Export Citation Format

Share Document