On the mixed boundary-value problem for harmonic functions in plane domains

This paper considers the problem of the determination of a harmonic function in a simply connected plane domain when the values of the function are known on some arcs of the boundary and the values of the normal derivative are known on the remaining boundary. We first present the solution in theoretical form and then show how to compute the solution with the aid of rapidly convergent series. The coefficients of these series are Fourier coefficients of certain functions and can be estimated by using the Fast Fourier Transform. The examples considered in the last part of the paper emphasize the advantages of the method presented in this paper for solving the mixed boundary-value problem as compared to other methods used for this purpose.

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Résumé On considère le problème suivant: trouver une fonction harmonique dans un domaine simplement connexeDR ² dont la frontièreS est assez régulière, en connaissant les valeurs de la fonction sur une partieS ₁ de la frontière et les valeurs de la dérivée normale sur la complémentaire de366-1. La première partie présente des résultats théoriques liés à ce probléme. La deuxième partie s'appuie sur une représentation de la solution à l'aide de certaines séries de fonctions rapidement convergentes. Les coefficients des séries utilisées sont même les coefficients de Fourier de certaines fonctions, et leur calcul peut être effectué en utilisant la transformation de Fourier rapide (Fast Fourier Transform). Les exemples considérés dans la dernière partie de l'article mettent en évidence les avantages de la méthode présentée par rapport à d'autres méthodes employées pour la résolution du problème mixte.

     

Quadrature rules for rational functions

Summary. It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turán, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. Received June 21, 1999 / Revised version received September 14, 1999 / Published online June 21, 2000     

Nodal domains and growth of harmonic functions on noncompact manifolds

Harmonic functions are studied on complete Riemannian manifolds. A decay estimate is given for bounded harmonic functions of variable sign. For unbounded harmonic functions of variable sign, relations are derived between growth properties and nodal domains. On Riemannian manifolds of nonnegative Ricci curvature, it has been conjectured that harmonic functions, having at most a given order of polynomial growth, must form a finite dimensional vector space. This conjecture is established in certain special cases.     

On boundary behavior of harmonic functions in Hölder domains

We analyze the boundary behavior of harmonic functions in a domain whose boundary is locally given by a graph of a Hölder continuous function. In particular we give a non-probabilistic proof of a Harnack-type principle, due to Bañuelos et al. and study some properties of the harmonic measure.     

Quadrature rules for integrals of fuzzy-number-valued functions

In this paper, we introduce some quadrature rules for the Henstock integral of fuzzy-number-valued mappings by giving error bounds for mappings of bounded variation and of Lipschitz type. We also consider generalizations of classical quadrature rules, such as midpoint-type, trapezoidal and three-point-type quadrature. Finally, we study δ-fine quadrature rules and we present some numerical applications.     

Quadrature domains and kernel function zipping

It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be “zipped” down to a strikingly small data set. It is also proved that the kernel functions associated to a quadrature domain must be algebraic. Research supported by NSF grant DMS-0305958.     

Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

In this paper we prove new results for p harmonic functions, p≠2, 1<p<∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p,n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p≠2, 1<p<∞, of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p=2).     

Quadrature and orthogonal rational functions

Classical interpolatory or Gaussian quadrature formulas are exact on sets of polynomials. The Szegő quadrature formulas are the analogs for quadrature on the complex unit circle. Here the formulas are exact on sets of Laurent polynomials. In this paper we consider generalizations of these ideas, where the (Laurent) polynomials are replaced by rational functions that have prescribed poles. These quadrature formulas are closely related to certain multipoint rational approximants of Cauchy or Riesz–Herglotz transforms of a (positive or general complex) measure. We consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence.     

Quadrature rule-based bounds for functions of adjacency matrices

Bounds for entries of matrix functions based on Gauss-type quadrature rules are applied to adjacency matrices associated with graphs. This technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities (Estrada index, subgraph centrality, communicability) that describe properties of networks.     

Uniqueness theorem for harmonic functions

If the boundary values of a function, harmonic in a sphere, and its normal derivative decrease sufficiently fast to zero as a fixed point of the sphere is approached, then the corresponding function is identically zero. This note gives an unimprovable condition on the rate of decrease for which the stated uniqueness theorem holds.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 247–252, March, 1968.     

Ostrowski-type theorems for harmonic functions

Ostrowski showed that there are intimate connections between the gap structure of a Taylor series and the behaviour of its partial sums outside the disk of convergence. This paper investigates the corresponding problem for the homogeneous polynomial expansion of a harmonic function. The results for harmonic functions display new features in the case of higher dimensions.     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

Quadrature formulae with free nodes for periodic functions

Summary. The problem of existence and uniqueness of a quadrature formula with maximal trignonometric degree of precision for 2-periodic functions with fixed number of free nodes of fixed different multiplicities at each node is considered. Our approach is based on some properties of the topological degree of a mapping with respect to an open bounded set and a given point. The explicit expression for the quadrature formulae with maximal trignometric degree of precision in the 2-periodic case of multiplicities is obtained. An error analysis for the quadrature with maximal trigonometric degree of precision is given. Received April 16, 1992/Revised version received June 21, 1993