On a generalized Carleman problem for a simply connected domain

1976 ◽  
Vol 27 (4) ◽  
pp. 456-458
Author(s):  
V. I. Slizkii
Keyword(s):  
Connected Domain ◽  
Simply Connected Domain ◽  
Carleman Problem ◽  
Simply Connected

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.

scholarly journals On harmonic continuation

1963 ◽  
Vol 6 (1) ◽  
pp. 54-56
Author(s):  
M. S. P. Eastham
Keyword(s):  
Analytic Function ◽  
Finite Number ◽  
Connected Domain ◽  
Arc Length ◽  
Jordan Curves ◽  
Simply Connected Domain ◽  
Harmonic Continuation ◽  
Simply Connected

Let D be a bounded, closed, simply-connected domain whose boundary C consists of a finite number of analytic Jordan curves. Let γ be any analytic arc of C. Then we shall prove the following theorem.Theorem 1. Let u(x, y) be harmonic in the interior of D and continuous on γ, and let ϱu(x, y)/ϱn=g(s) when (x, y) is on γ, where g(s) is an analytic function of arc-length s along γ. Then u(x, y) can be harmonically continued across γ.

Some remarks on universal functions and Taylor series

2000 ◽  
Vol 128 (1) ◽  
pp. 157-175 ◽  
Author(s):  
G. COSTAKIS
Keyword(s):  
Complex Plane ◽  
Taylor Series ◽  
Connected Domain ◽  
Constructive Proof ◽  
Universal Functions ◽  
Baire’S Theorem ◽  
Simply Connected Domain ◽  
Universal Taylor Series ◽  
Simply Connected

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.

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

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