Introduction to Potential Theory. By L. L. Helms. Pp. ix, 282. £7. 1970. (Wiley.)

The present state as well as the historical development of problems related to the mean value property of harmonic and caloric functions are discussed. The article is based on contributions of the authors at the meeting; it is a substantially extended update of [Net2].

Mean Value Property and Harmonic Functions

We consider several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here.

Duality theorems in some overdetermined boundary value problems

Publisher Summary This chapter presents a bibliography of multivariate approximation. Multivariate interpolation and approximation have always been an important part of general Approximation Theory, however, in the past ten years or so, there has been a significant increased interest in multivariate problems. A very substantial number of both theoretical and practical results have been obtained on a variety of topics, ranging from the classical tensor-product methods to the newfangled multivariate splines. In addition to their applicability to such classical problems as data fitting and numerical solution of operator equations, modern multivariate methods have become important in several new applied areas, such as computer-aided design, robotics, and computer-aided tomography.

A bibliography of multivariate approximation

Let D be an open subset of RN N ≧ 2, such that O ∊ D and λ(D) < + ∞, where λ denotes N-dimensional Lebesgue measure. If the mean-value equality holds for every integrable harmonic function h on D. then accoeding to a theorem of KuranD is a ball of centre O. Here we show that the same conclusion holds if we assume the above mean-value equality only for positive integrable harmonic functions. Further, if the mean-value equality is assumed only for bounded harmonic functions, then D = B=\E, when B is an open ball and E is a polar set.

The volume mean-value property of harmonic functions

Etude de la propriete de la moyenne des fonctions harmoniques. Existence, unicite et caracterisation de la meilleure approximation L 1 harmonique des fonctions strictement sous harmoniques

On the mean value property of harmonic functions and best harmonic ¹-approximation

Blending interpolation and bestL 1-approximation

Burchard [B] and Hayman, Kershaw and Lyons [HKL] have considered uniform approximation by harmonic functions, and their results have been generalized to deal with approximation by Solutions of elliptic partial diiferential equations by Haussmann and Zeller [HZ] s well s Gauthier and Zwick [GZ]. Also, Zwick [Z] and Wilson and Zwick [WZ] have dealt with uniform approximation by superharmonic functions.

Characterization of best harmonic and superharmonic L1-approximations.

On the inverse mean value property of harmonic functions on strips

Let \(\), where B is the open unit ball in\(\) (\(\)), and let \(\) denote the collection of functions h in \(\) which are harmonic on B and satisfy \(\) on \(\). A function h * in \(\) is called a best harmonic one-sided L 1-approximant to f if \(\) for all h in \(\). This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation on the cylinder \(\) are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harmonic approximation. The characterizations are quite different in nature from those recently obtained for harmonic L 1-approximation without a one-sidedness condition.

Werner Haussmann

Papers

On the mean value property of harmonic functions and best harmonic ¹-approximation

Blending interpolation and bestL 1-approximation

Characterization of best harmonic and superharmonic L1-approximations.

On the inverse mean value property of harmonic functions on strips

Best one-sided L 1-approximation by harmonic functions