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Search: MSC category 31B20 ( Boundary value and inverse problems )

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1. CMB Online first

Khavinson, Dmitry; Lundberg, Erik; Render, Hermann
The Dirichlet problem for the slab with entire data and a difference equation for harmonic functions
It is shown that the Dirichlet problem for the slab $(a,b) \times \mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$ the inhomogeneous difference equation $h ( t+1,y) -h (t,y) =g ( t,y)$ has an entire harmonic solution $h$.

Keywords:reflection principle, entire harmonic function, analytic continuation
Categories:31B20, 31B05

2. CMB 2003 (vol 46 pp. 252)

Miyamoto, Ikuko; Yanagishita, Minoru; Yoshida, Hidenobu
Beurling-Dahlberg-Sjögren Type Theorems for Minimally Thin Sets in a Cone
This paper shows that some characterizations of minimally thin sets connected with a domain having smooth boundary and a half-space in particular also hold for the minimally thin sets at a corner point of a special domain with corners, {\it i.e.}, the minimally thin set at $\infty$ of a cone.

Categories:31B05, 31B20

3. CMB 1997 (vol 40 pp. 60)

Khavinson, Dmitry
Cauchy's problem for harmonic functions with entire data on a sphere
We give an elementary potential-theoretic proof of a theorem of G.~Johnsson: all solutions of Cauchy's problems for the Laplace equations with an entire data on a sphere extend harmonically to the whole space ${\bf R}^N$ except, perhaps, for the center of the sphere.

Keywords:harmonic functions, Cauchy's problem, homogeneous harmonics
Categories:35B60, 31B20

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