Alternative derivations for the Poisson integral formula

Poisson integral formula is revisited. The kernel in the Poisson integral formula can be derived in a series form through the direct BEM free of the concept of image point by using the null-field integral equation in conjunction with the degenerate kernels. The degenerate kernels for the closed-form Green's function and the series form of Poisson integral formula are also derived. Two and three-dimensional cases are considered. Also, interior and exterior problems are both solved. Even though the image concept is required, the location of image point can be determined straightforward through the degenerate kernels instead of the method of reciprocal radii.     

Extreme harmonic functions on a ball

In order to show that one can recapture the Riesz-Herglotz theorem from the Krein-Milman theorem, we determine directly the set of extreme points of the convex set of positive harmonic functions on the unit ball (normalized by 1 at the origin). The characterization is obtained using standard facts from abstract analysis combined with a minimum of very basic results on harmonic functions.     

Harmonic wavelets in boundary value problems for harmonic and biharmonic functions

We consider boundary value problems in a disk and in a ring for homogeneous equations with the Laplace operator of the first and second orders. Solutions are represented in terms of bases of harmonic wavelets in Hardy spaces of harmonic functions in a disk and in a ring, which were constructed earlier.     

Optimal estimates for harmonic functions in the unit ball

We find the sharp constants C _p and the sharp functions C _p ?=?C _p (x) in the inequality $$ |u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}} \|u\|_{h^{p}(B^{n})}, u\in h^{p}(B^{n}), x\in B^{n}, $$ in terms of Gauss hypergeometric and Euler functions. This extends and improves some results of Axler et?al. (Harmonic function theory, New York, 1992), where they obtained similar results which are sharp only in the cases p = 2 and p = 1.     

Sharp integral inequalities for harmonic functions

Motivated by Carleman's proof of isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper half‐space. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville‐type theorems. © 2007 Wiley Periodicals, Inc.     

Approximation of conjugate differentiable functions by biharmonic Poisson integrals

We determine the exact values of upper bounds of approximations by biharmonic Poisson integrals on classes of conjugate differentiable functions in uniform and integral metrics.     

A monotonicity formula for stationary biharmonic maps

We give a rigorous proof of the monotonicity formula of S.-Y.A. Chang, L. Wang and P. Yang [3] for (extrinsically) stationary biharmonic maps of class W^2,2. This work was partially supported by SNF 200021-101930/1.     

Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L ^p . This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L ^p -norm of its boundary values is attained at the radial direction. This confirms D. Khavinson’s conjecture for p = 1 and p = 2. Bibliography: 11 titles.     

A generalization of a formula of Steiner

Summary Combinatorial formulae for the numbers of cells of subdivision, of varying dimensions, when hyperplanes in general position intersect an open convex set of ^d, are derived. They coincide with basic formulae for the probabilities of combinations of events. The dual result is stated.I should like to thank a referee for drawing attention to the duality.     

On translates of the Poisson kernel and zeros of harmonic functions

We characterize the systems of translates of the Poisson kernelspanning L^p(), 1 p < . An equivalent formulation in termsof discrete uniqueness sets for harmonic functions is given,together with a Blaschke-type condition for the zero varietyof bounded harmonic functions in the unit disk.     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.     

A generalization of the Christoffel-Schwartz formula

As a generalization of the well-known Christoffel-Schwartz formula, a formula is obtained for mapping the interior of the unit disk onto a domain whose boundary consists of n arcs of curves, each of which, under the choice of some branch of the transformation ζ=w^m, m>0, passes through a rectilinear segment of the ζ-plane. It is shown that the class B_m of Bazilevich functions coincides with the class ¯L_m of functions representable by means of the obtained formula of the special type.     

A generalization of the denjoy integral

We prove that the IsekiM-integral is substantially more general than the Denjoy integral. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 766–772, November, 1997. Translated by I. P. Zvyagin