Mixed boundary value problems for concentric cylinders—an algebraic operator formulation

Unitary operators are introduced which act on the Fourier transforms of the boundary values on concentric cylinders of a harmonic function in E ₃. The boundary values on the concentric cylinders are determined from given mixed boundary conditions on the same cylinders, and the solution for them is expressed explicitly and simply through the use of these unitary operators plus certain other self adjoint positive operators. The boundary values on the cylinders then determine the harmonic function everywhere in E ₃.     

Circle boundaries of planar graphs

Letg be an infinite, connected, planar graph with bounded vertex degree, which obeys a strong isoperimetric inequality and which can be embedded in the plane so that each cycle surrounds only finitely many vertices. We investigate a certain class of compactifications ofg; one of which has boundary homemorophic to a circle. We shall show that ifg is a tree or, more generally, ifg is hyperbolic, then this circle boundary supports an integral representation of any given bounded harmonic function. We further show that in the specific case of a triangulation of the plane, the graph is hyperbolic and therefore the Martin boundary is a circle.     

Local existence and uniqueness theorems for a free boundary problem

Free boundary problems are considered, where the tangential and normal components u_t and u_n of an otherwise unknown plane harmonic vector field are prescribed along the unknown boundary curve as a function of the coordinates x, y and the tangent angle θ. The vector field is required to exist either in the interior region G⁺ or in the exterior G^?. In each case the free boundary is characterized by a nonlinear integral equation. A linearised version of this equation is a one-dimensional singular integral equation. Under rather general hypotheses which are easy to check, the properties of the linear equation are described by Noether's theorems. The regularity of the solution is studied and the effect of the nonlinear terms is estimated. A variant of the Nash-Moser implicit-function theorem can be applied. This yields local existence and uniqueness theorems for the free boundary problem in Hölder-classes H^2+μ. The boundary curve depends continuously on the defining data. Finally some examples are given, where the linearised equation can be completely discussed.     

Sharp real-part theorems in the upper halfplane and similar estimates for harmonic functions

Explicit formulas for sharp coefficients in estimates of the modulus of an analytic function and its derivative in the upper half-plane are found. It is assumed that the boundary values of the real part of the function are in L ^p . As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in \mathbb R₊² {\mathbb R}_{+}^2 are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in \mathbb R₊ⁿ {\mathbb R}_{+}^n by the L ^p -norm of the boundary normal derivative is given, 1 \leqslant p \leqslant ￥ 1 \leqslant p \leqslant \infty . This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p ∈ [1, n]. Bibliography: 6 titles.     

The Dirichlet problem of a discontinuous Markov process

Given a Markov process satisfying certain general type conditions, whose paths are not assumed to be continuous. LetD be an open subset of the state spaceE. Any bounded function defined on the complement ofD extends to be a function onE such that it is harmonic inD and satisfies the Dirichlet boundary condition at any regular boundary point ofD. The relation between harmonic functions and the characteristic operator of the given process is discussed.     

Quasiconformal harmonic mappings between \fancyscriptC1,m{{\fancyscript{C}}^{1,\mu}} Euclidean surfaces

The conformal deformations are contained in two classes of mappings quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every K quasiconformal harmonic mapping between surfaces with boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

Asymptotics of small spacings for subsonic nonstationary streamlined flow of a thin airfoil close to a solid boundary

In an asymptotic approximation of small spacings an analytic solution to the problem on harmonic oscillations of a thin airfoil which is moving with a subsonic velocity near a solid plane boundary is given. Results of a computation of the lifting force are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 48–53, 1988.     

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R_m (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V_{3, 3}). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R_m (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.     

Finely locally injective finely harmonic morphisms

We consider fine topology in the complex plane C and finely harmonic morphisms. We use oriented Jordan curves in the plane to prove that for a finely locally injective finely harmonic morphism f in a fine domain in C, either f or f is a finely holomorphic function. This partially extends result by Fuglede, who considered a kind of continuity for the fine derivatives of the finely harmonic morphism. As a consequence of this we obtain a both necessary and sufficient condition for a function f to be finely holomorphic or finely antiholomorphic. We do not know if the condition of finely local injectivity (q.e.) is automatically fulfilled by any non-constant finely harmonic morphism.     

Domain identification for harmonic functions

The authors investigate the problem of identifying the domainG of a harmonic functionu such that Cauchy data are given on a known portion of the boundary ofG, while a zero Dirichlet condition is specified on the remaining portion of the boundary, which is to be found. Under certain conditions on the domainG, it is shown that the problem reduces to identifying the coefficients of an elliptic equation which, in turn, is converted into the problem of minimizing a functional. Under certain conditions onG, it is shown that the solution, if it exists, is unique. An application is pointed out for the problem of designing a vessel shape that realizes a given plasma shape.This work was completed with a financial support from the National Basic Research in the Natural Sciences.     

插值逼近具边界数据的调和函数

该文考虑光滑闭Jondan曲线Γ围成的单连区域D,证明了在Γ上具有已知导数数据的D内调和函数u(x,y)的存在性.继而构造了一个调和插值多项式序列在(?)=D∪Γ上一致收敛于u(x,y),且具理想的收敛速度.此外,以往同类研究工作中的边界Γ是解析曲线,而在该文中已减少边界限制为Γ∈J_0.     

Spectral factorization in the disk algebra

Every strictly positive function f, given on the unit circle of the complex plane, defines an outer function. This article investigates the behavior of these outer functions on the boundary of the unit disk. It is shown that even if the given function f on the boundary is continuous, the corresponding outer function is generally not continuous on the closure of the unit disk. Moreover, any subset E∈ [-π ,π) of Lebesgue measure zero is a valid divergence set for outer functions of some continuous functions f. These results are applied to study the solutions of non-linear boundary-value problems and the factorization of spectral density functions.     

The harmonic dirichlet problem for a cracked domain with jump conditions on cracks

The harmonic problem in a cracked domain is studied in R ^m , m?>?2. The boundary of the domain is assumed to be nonsmooth, while cracks are smooth. The Dirichlet condition is specified on the boundary of the domain. Jumps of the unknown function and its normal derivative are specified on the cracks. Uniqueness and solvability results are obtained. The problem is reduced to the uniquely solvable integral equation, its solution is given explicitely in the form of a series. The estimates of the solution of the problem depending on the boundary data are obtained.     

Modulus of continuity of harmonic functions

LetG be a bounded plane domain, the diameters of whose boundary components have a fixed positive lower bound. Letu be harmonic inG and continuous in the closureG ofG. Suppose that the modulus of continuity ofu on the boundary ofG is majorized by a function of a suitable type. We shall then obtain upper bounds for the modulus of continuity ofu inG. Further, we shall show that in some situations these estimates cannot be essentially improved. We shall also consider the same problem for certain bounded domains in space. Research partially supported by the U.S. National Science Foundation. AMS (1980) Classification. Primary 31A05.     

Representations of bounded harmonic functions

Summary An open subsetD ofR ^d ,d≧2, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a GreenianD to be Poissonian in terms of its Martin boundary. Supported by NSF DMS86-01800.     

A generalization of the H-theorem to steady non-equilibrium states. Part 1: A basic decomposition and the linear case

A generalized relative entropy functional is associated to the evolution of gas in a container with (generally) non-uniform boundary data. A decomposition for its rate in ‘bulk’ and in ‘boundary’ terms is given; for the linear case both have a definite sign. The relation with the strong L₁ stability is pointed out.     

Integral representations in Hardy classes and best approximations in certain functional spaces

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H¹ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 342–347, March, 1991.     

Integral representations in hardy classes and best approximations in certain functional spaces

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class H¹ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 342–347, March, 1991.     

Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations

We show that theL ^p norms, 0<p<∞, of the nontangenital maximal function and area integral of solutions and normalized adjoint solutions to second order nondivergence form elliptic equations, are comparable when integrated on the boundary of a Lipschitz domain with respect to measures, which are respectivelyA _∞ with respect to the corresponding harmonic measure or normalized harmonic measure. Both authors are supported by NSF