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## Über dieses Buch

Within the tradition of meetings devoted to potential theory, a conference on potential theory took place in Prague on 19-24, July 1987. The Conference was organized by the Faculty of Mathematics and Physics, Charles University, with the collaboration of the Institute of Mathematics, Czechoslovak Academy of Sciences, the Department of Mathematics, Czech University of Technology, the Union of Czechoslovak Mathematicians and Physicists, the Czechoslovak Scientific and Technical Society, and supported by IMU. During the Conference, 69 scientific communications from different branches of potential theory were presented; the majority of them are in­ cluded in the present volume. (Papers based on survey lectures delivered at the Conference, its program as well as a collection of problems from potential theory will appear in a special volume of the Lecture Notes Series published by Springer-Verlag). Topics of these communications truly reflect the vast scope of contemporary potential theory. Some contributions deal with applications in physics and engineering, other concern potential­ theoretic aspects of function theory and complex analysis. Numerous papers are devoted to the theory of partial differential equations. Included are also many articles on axiomatic and abstract potential theory with its relations to probability theory. The present volume may thus be of intrest to mathematicians speciali­ zing in the above-mentioned fields and also to everybody interested in the present state of potential theory as a whole.

## Inhaltsverzeichnis

### On Weighted Beppo Levi Functions

Let 1 < p < ∞ and let w be a weight satisfying the Muckenhoupt AP condition. We write $$\left\| f \right\|_{L^p ,w} = (\int\limits_{R^n } {\left| {f(x)} \right|} ^p w(x)dx^{1/p} ,L^p (R^n ,w) = \{ f;\left\| f \right\|_{L^p ,w} < \infty$$.

Hiroaki Aikawa

### Generalized Cauchy-Riemann Equations and the Positive -Subharmonic Functions

In this paper we will give a statement about the positive solutions of the generalized Cauchy-Riemann system in R2n. It will also be shown that the positive solutions are related with the “$$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{k}$$-subharmonicity”.

Ömer Akin, A. Okay Çelebi

### Iteration Methods for Potential Problems

Two iteration methods are considered as a means of constructing solutions of various integral equation formulations of the exterior Dirichlet and Neumann potential problems. These integral equations may be of the second kind, in which case the integral operators will in general be non self-adjoint, or of the first kind, which in the Dirichlet case leads to self-adjoint operators.

Thomas S. Angell, Ralph E. Kleinman, Gary F. Roach

### On the Poisson Equation on the Infinite Dimensional Torus

Let {μt}t>0 be the Wiener semigroup on the infinite dimensional torus T∞ (cf. Heyer,5 Ch. 5). Let us consider the Green measure $$G(dy) = \int\limits_0^\infty {e^{ - t} \mu _t (dy)dt}$$.

Alexander D. Bendikov, Igor V. Pavlov

### A Characterization of Elementary Potential Theory

In the following (Vλ) will be a Ray resolvent on a compact metric space E with set D of non-branchpoints, E denotes the class of positive Borel functions on E, and S is the cone of functions on D, which are excessive for (Vλ). For simplification we assume V1:= V01 to be continuous.

### Wiener Estimates for Solutions of Elliptic Equations in Nondivergence Form

In this paper we deal with a Wiener estimate of the modulus of continuity at the boundary for the solutions of elliptic problems in nondivergence form. It is well known that the Wiener criterion for the continuity of the Laplace equation at a point of the boundary was given at first by Wiener (1923) and was extended to general linear elliptic (second order) equations in divergence form by Littman, Stampacchia, Weinberger (1963) (for the intermediate stages see the references of this last paper).

Marco Biroli

### Approximation by Continuous Potentials

In this note we improve theorems in [1] and [2] dealing with approximation of (super)harmonic functions by continuous potentials. That is, we intend to show that for every finely open set G of a balayage space (X, W) there exists a continuous potential q ε P such that $$S(G) = \overline {P + \mathbb{R}q} ,H(G) = \overline {H(q)}$$.

Jürgen Bliedtner, Wolfhard Hansen

### The Schrödinger Equation Δu = µu in a Neighbourhood of an Isolated Singularity of µ

In this paper we present new results about the Picard principle and the behaviour of the continuous solutions of the Schrödinger equation Δu = µu at an isolated singularity of the measure µ. Using the axiomatic potential theory, a criterion for the integrability of µ is given.

Abderrahman Boukricha

### A Characterization of the Fine Sheaf Property

Throughout this article S will be a standard H-cone of functions on a set X. For definitions and notations see Springer Lecture Notes in Mathematics 853.

Aurel Cornea

### Harmonic Morphisms and Ray Processes

In the following we will let (X, U) and (Y, V) denote P-harmonic spaces in the sense of Constantinescu and Cornea [3]. (X, Y are locally compact Hausdorff spaces with countable bases and U, V are hyperharmonic sheaves on respectively.) We will assume that the constant function 1 is hyperharmonic. It is now well known that there exist Hunt processes Xt,Yt on X, V with continuous paths (i.e. diffusions) such that the family of Xt-excessive, resp. Yt-excessive, functions coincide with the family of non-negative U-hyperharmonic, resp. V-hyperharmonic, functions.

Laszlo Csink, Bernt Øksendal

### On Dirichlet’s Boundary Value Problem for Certain Anisotropic Differential and Pseudo-Differential Operators

In our previous paper [1] a calculus for pseudo-differential operators was used to find periodic solutions for a large class of not necessarily elliptic pseudo-differential equations with constant coefficients. In this paper we consider a generalized homogeneous Dirichlet-problem for pseudo — differential operators with constant coefficients and prove Fredholm’s alternative theorem. We give two examples how to apply this result. The first one is a differential operator arising in stochastics. In the second example we deal with an anisotropic pseudo-differential operator and using the theory of Dirichlet forms (see [4]) it follows that this operator generates a symmetric Hunt process. Some of our results had been already used in [6].

Karl Doppel, Niels Jacob

### Hyperharmonic Cones

The theory of H-cones ([3]) covers the superharmonic case in potential theory. This work continues the algebraic axiomatization of the hyperharmonic case. Our basic structure is a hyperharmonic structure defined by M. Arsove and H. Leutwiler in [1]. The cancellation law does not hold in hyperharmonic structures. In order to characterize cancellable elements we assume that a hyperharmonic structure is a convex cone and the greatest lower bound of (u/n) nεℕ exists for all u.

Sirkka-Liisa Eriksson-Bique

### Jump Relations for Surface Potentials in Case of Continuous Densities

It is well-known that the single layer surface potential with uniform HÖLDER continuous density is continuously differentiable with respect to each side of the surface and fulfills jump relations for the first order derivatives. In this paper it is shown that the jump relations still hold in the CAUCHY sense if the density is supposed to be only continuous.

Alexander Hornberg

### On a Unified Characterization of Capacity

The notion of capacity as a fundamental quantity associated with a smooth surface was treated exhaustively by Polya and Szego in a series of papers over more than twenty years following earlier work by Fekete (1923) and many others who considered the problem of determining the charge accumulated on an electrostatic condenser maintained at a constant potential. See Polya and Szego (1951) and Payne (1967) for extensive references.

George C. Hsiao, Ralph E. Kleinman

### A Note on the Computation of Green Functions for Sublaplacians on Lie Algebras of Type H

In [4] A. Kaplan introduced a class of step-2 nilpotent Lie groups for which the fundamental solution corresponding to the standard sublaplacian may be written down in an explicit elementary form. He also says that heuristic evidence suggests that these groups are the only ones which admit such nice explicit formulas. When I tried to convince myself of this evidence I found some modifications of Kaplan’s proof, which give additional insight into the problem and may be — at least for people working in potential theory — regarded as a simplification.

H. Hueber

### On Thermoelastic Potential

S.Klainerman, in [1] proved that for suficiently small nonlinear perturbations of linear dissipative equations and systems like: the wave equation, heat equation, linear isotropic elasticity equations etc., the corresponding solution of the initial-value problem for perturbed equations and system behave asymptotically like the solutions of the initial-value problem for linear unperturbed equations and system.

Hung Do Duc, Jerzy Gawinecki, Adam Piskorek

### The Dirichlet Problem on Ends

In the recent investigations of the Dirichlet problem in an axiomatic framework we focus our interest on the following two problems: (1) the boundary behavior of the normalized PWB-solutions, originally studied by O. Frostman[3] and generalized by Constantinescu-Cornea[2] and Lukeš-Malý[11] and (2) the unicity of Keldych operators formulated by J. Lukeš[10], given by Bliedtner-Hansen[1] its essential contribution. The normalized PWB-solutions HfU,X on an open set U is formed by the Perron-Wiener-Brelot’s method taking the closure of U in the one-point compactified space and putting f to be 0 at the infinity, and the effect of the ideal boundary is neglected. In this article we consider above problem under the full influence of the ideal boundary and in the conclusing result we reflect that our result is a generalized version of that obtained formerly.

Teruo Ikegami

### Diffusion Kernels of Logarithmic Type

Let X be a locally compact, non-compact Hausdorff space with countable basis. We denote by: CK(X) the usual topological vector space of all finite continuous functions with compact support;C(X) the usual Fréchet space of all finite continuous functions on X;MK(X) the usual topological vector space of all real Radon measures with compact support;M(X) the topological vector space of real Radon measures on X with the weak topology.

Masayuki ItÔ

### A Remark on Capacitary Measures for Diffusion Processes

In this note we shall give a characterization of capacitary measures of semipolar sets for diffusion processes, which is closely related to Dellacherie, Feyel and Mokobodzki2. In this study, the process X = (Xt) is assumed to be a standard process on (E, E), where E is a locally compact separable Hausdorff space and E is the Borel algebra on E. For X we suppose the following: i) X satisfies the strong duality hypotheses relative to a σ-finite measure ξ(dx) = dx; ii) U(x, K) and Û(x, K) are bounded on E for each compact subset K of E; iii) the kernel density u(x, y) of U(x, dy) relative to ξ is continuous off the diagonal and x → u(x, y), y → u(x, y) are lower semicontinuous. The brief explanation of the notations above will be given in the subsequent section. But we assume that the reader is familiar with terminologies in Blumenthal and Getoor1. A process is called a diffusion process on E if it is a standard process on (E, E) with continuous paths. For a subset B of E, TB denotes the hitting time of B, that is, TB = inf(t>0, XtεB).

Mamoru Kanda

### On the Convexity of Level Sets for Elliptic and Parabolic Exterior Boundary Value Problems

The talk is divided into two parts. First I study an elliptic free boundary problem and show that sometimes it is useful to work with curvilinear coordinates. Then I present some results on parabolic problems, which I obtained in cooperation with I. Diaz.

Bernhard Kawohl

### Harnack Inequality and Some Characteristics of Non-Regular Points of Simplexes

Let S be a compact convex set, E=E(S) — the set of all extreme points of S, Shs=Ē\E — the set of all non-regular points of S; C(Ē) — the vector space of all real continuous functions on Ē.

D. G. Keselman

### Polar Sets in a Nonlinear Potential Theory

In this lecture we discuss nonlinear potential theory based on “A-super-harmonic functions”; the theory can be viewed as a (nonlinear) extension of the classical study of superharmonic functions in ℝn.

T. Kilpeläinen

### A Sharper Form of a Theorem of Kolmogorov

1°. Let H+ be the class of functions f(z) holomorphic in $$\mathbb{D} = \left\{ {z \in \mathbb{C}:\left| z \right| < 1} \right\}$$ and such that Ref(z)>0. Functions in H+ have the representation (1)$$\int_{D} {f(w)dudv = L(f)} (w = u + iv)$$

Boris Korenblum

### Extremal Problems for the Distributed Moment Problem

The present work continues the investigation of the socalled distributed moment problem (DMP) (see [13] and references there). We use the last name instead of its alternative one, inverse potential problem [19], to stress the analogy with the one-dimensional moment problem [1,8,15].

Ognyan Kounchev

### On Mathematical Modelling of Non-Newtonian Flow by Perturbation Methods

Isothermal incompressible flow in consideration of friction influences can generally be described by the non-linear system of partial differential equations (1)$$\text{Re}\,\frac{{\text{dv}}}{{\text{dt}}}\, = \, - \operatorname{Re} \,.\,\text{grad}\,\text{p}\,\text{ + }\,\text{DIV}\,\text{T}\,\text{ + }\,\text{f}\,\text{,}\,\text{div v}\,\text{ = }\,\text{0}\text{.}$$

R. Kreul, H. Kretzschmar

### On Diffusion Semigroups Generated by Semi-Elliptic Differential Operators in Infinite Dimensions

We want to study some continuity properties of operator semigroups, generated by a semi-elliptic differential operator on a real separable Hilbert space ℍ. To this end, let us begin by writing the finite-dimensional semi-elliptic differential operator (1)$$\text{Lu(x) = }\frac{\text{1}}{\text{2}}\sum\limits_{i,j = 1}^n {a_{ij} } (x)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}(x) + \sum\limits_{i = 1}^n {b_i } (x)\frac{{\partial u}}{{\partial x_i }}(x)$$ on ℍ = ℝn in coordinate-free form as (1a)$$Lu(x) = \frac{1}{2} tr u''(x)(a(x)\cdot ,\cdot ) + u'(x)(b(x))$$

Gottlieb Leha

### The Obstacle Problem in a Non-Linear Potential Theory

M. Brelot gave rise to the concept harmonic space when he extended classical potential theory on ℝn to an axiomatic system on a locally compact space. I have recently constructed1 a non-linear harmonic space by dropping the assumption that the sum of two harmonic functions is harmonic and considering some other axioms instead. This approach has its origin in the work of O. Martio, P. Lindqvist and S. Granlund2,3,4, who have developed a non-linear potential theory on ℝn connected with variational integrals of the type ∫ F(x,∇u(x)) dm(x), where F(x, h) ≈ |h|p.

P. Lehtola

### On the Appell Transformation

The main purpose of this paper is to show that the well known Appell transformation [1] is essentially — i.e. up to compositions with the obvious transformations of the form (1.2) mentioned below — the only transformation, mapping one solution of the n-dimensional heat equation $$\Delta u\, = \,\frac{{\partial u}}{{\partial t}}$$ into another (see Corollary 2.2 for a precise statement).

Heinz Leutwiler

### Perturbation and Excessive Functions

Originally, N.Boboc and Gh.Bucur [2] studied perturbations of a resolvent or a semigroup on a measurable space, by bounded kernels. They gave a characterization of the perturbed excessive structures.

Habib Maagli

### Capacities on Harmonic Spaces with Adjoint Structure

In the classical potential theory, the capacities defined in terms of Green potentials coincide with the capacity defined by Dirichlet integrals; more precisely, for a compact set K in a Greenian domain Ω in ℝd, $$Sup\{ \mu (\Omega )|G\mu \leqq 1,\,Supp\,\mu \subset K\} = \inf \{ \smallint G\mu \,d\mu |G\mu \geqq 1\,on\,K\} = \,\inf \{ D[f]\,|\,f:\,potential\,on\,\Omega \,with\,f \geqq 1\,on\,K\}$$, where Gμ is the Green potential of μ ≧ 0 on Ω and D[f] is the Dirichlet integral of f.

Fumi-Yuki Maeda

### Balayage Spaces on Topological Sums

Several authors, e.g. N. Bouleau /6/, N. Boboc and Gh. Bucur /4/, H. Ben Saad and K. Janßen /2/, A. Boukricha /5/, studied potentialtheoretic structures on topological sums and their connection uith the biharmonic spaces of E.P. Smyrnélis /10/.

Michael Meyer

### Subordination for Balayage Spaces

Let X be a Iccb space, Uand W two sub-Markov resolvents on X possessing proper potential kernels U and V such that W is subordinated to U (i. e. Vλ ≤ Uλ Vλ]0, ∞[). Denote by S U , S W respectively E U E` W the cone of super-median respectively excessive functions associated with U and W. The purpose of this article is: (i) to study the situation where (X, E U ) and (X, E W ) are both balayage spaces (in the sense of [3]) and P is the sub-Markov kernel on X satisfying U=V+PU, and (ii) to give conditions on U, V and P under which (X, E W ) respectively (X, E U ) form a balayage space, provided (X, E U ) respectively (X, E W ) is a balayage space. The notations are the notations of [3].

Hans-Helge Müller

### On the New Problem in the Potential Gravity Theory

The results presented in this paper were obtained in the course of an investigation of a simple model problem of the time-dependent Earth s gravity field, but they are equally applicable to other problems governed by thermo-magnetohydro-dynamics and geophysics. The author has studied the problem of the existence, uniqueness and regularity of the solution of the variational problem of the coupled gravity field, velocity field in thermo-Bingham’s fluid (by which the Earth s interior is simulated) and a quasi-stationary magnetic field.

Jiři Nedoma

### Stochastic Differentiation Formula and Convergence Theorem for Finely Hypoharmonic Functions

As a consequence of a convergence theorem for finely hyperharmonic functions given by us in [4e], we give here an elementary proof of the following fact: Let K ⊂ C be a compact set in the complex plane and let (fn) be a Cauchy sequence in (Cc(K)), sup | |) of functions holomorphic in the neighborhoods of K. Then for any point z in the fine interior K’ of k there is a compact fine neighborhood V of Z in K’ of K there is a compact fine neighborhood V of Z in K’ such that: The process (∂zfn(Zτ)) is locally a cauchy sequence in L2([0,∞[xΩ, μz,) for any Z’ in K’, where μz, denotes the Doléans measure on[0,+∞[xΩ associated with the complex standard Brownian motion started at z’ and stopped at the fine boundary of K’.

Nguyen Xuan-Loc

### Morphisms of Stonian Cones

The main result of this paper puts forward a class of morphisms of H-cones, possessing image: namely those commuting with arbitrary infimum. Applying this result to the canonical embedding in the bidual, one gets a decomposition of any H-cone In the particular case when the cone has the specific order, this decomposition gives a well-known theorem of Dixmier [1], on the structure of stonian compact spaces.

Eugen Popa

### On the Transition Function of the Infinite Dimensional Ornstein-Uhlenbeck Process Given by the Free Quantum Field

In this paper we want to investigate a particular semigroup (πt)t>0 of probability kernels defined on some infinite dimensional Banach space Bα contained in Y’(ℝd−1) (i.e. the tempered distributions on ℝd-1), d ≥ 2, $$\alpha \in ]\frac{{d - 2}}{2},\infty$$. This semigroup (πt)t>0 is the transition function of the infinite dimensional Ornstein-Uhlenbeck process given by the free field of Euclidean quantum field theory on ℝd (cf. [1], [2]). We will establish that (πt)t>0 has the Feller property, determine its generator (on a suitable domain) and its Dirichlet form. Furthermore, we will characterise the associated entrance space in the sense of Dynkin (cf. [3], [4], [5]).

Michael Röckner

### Finiteness of the Family of Simply Connected Quadrature Domains

A quadrature domain D of a functional L is a domain with finite area satisfying (1.1)$$\begin{array}{*{20}c}{\smallint _D f(w)dudv = L(f)} {(w = u + iv)} \\\end{array}$$ for every f in the class AL1(D) of functions analytic and integrable in D. A simple example is a disk: The unit disk is a quadrature domain of f ↦ πf(0). In this case, the quadrature domain is determined uniquely. But, in general, it is not easy to determine all quadrature domains of a given functional.

Makoto Sakai

### On an Inverse Problem for the Newtonian Potential

In a joint work with C. Maderna and C. Pagani,1 we have considered the following classical inverse problem of potential theory: to determine the figure of a Homogeneous material body G whose shape is unknown, from measurements of the Newtonian potential created by G, taken on the surface of a sphere containing G in its interior.

Sandro Salsa

### On the Integration of Elastostatic Displacement Equation

The Cauchy-Navier equation of motion for the linear elastic isotropic continuum, for the vanishing body forces, may be written in the form

Abdullah Shidfar

### Non-Linear Potential Theory in Lebesgue Spaces with Mixed Norm

The classical potential theory for the Riesz kernel in ℝd has been generalized in many different ways. In this paper we are concerned with the Lp-potential theory which appeared around 1970. See N.G. Meyers [Me] and V.G. Maz’ya and V.P. Havin [MH] for an introduction to this theory. Applications and further developments of the theory are found in the books V.G. Maz’ya [Ma], V.G. Maz’ya, T.O. Shaposhnikova [MS], the lecture notes D.R. Adams [A] and the survey article L.I. Hedberg [H]. See also [S1] and [S2].

Tord Sjödin

### Schroedinger Equations with Discontinuous Solutions

In this paper we study the Schrödinger equation $$( - \frac{1}{2}\Delta + \mu )h = 0$$ on a (possibly unbounded) domain U of Rd(d ≥ 1). µ will be a signed measure with µ+ ε Dloc and µ−ε D1. By definition, a measure v ≥ 0 is in the class Dloc if it is associated with an increasing continuous additive functional A v of standard Brownian motion (Px,X t : x ε Rd, t ≥ 0) such that for some t > 0 and every compact set C ⊂ Rd$$\mathop {\sup }\limits_{x \in R^d } E^x [A_v (t).1_c (X_t )] < \infty$$.

Theo Sturm

### Polynomial Hulls and Envelopes of Holomorphy

This is a report on recent joint work with Herbert Alexander.

John Wermer

### Green Function and Uniform Harnack Inequality

We show that any harmonic space which possesses a Green function with certain properties satisfies the uniform inequality. Also a lower bound for the harmonic measure of balls is given.

Rainer Wittmann

### On the Asymptotic Behavior of Solutions of a System of Integral Equations of Mixed Boundary Value Problem of Plane Elasticity in a Neighborhood of Corner Points of the Contour

In the papers [1–2] a method for investigation of asymptotics of solutions near singularities of the boundary of boundary integral equations, arising in problems of potential theory was proposed. This method is based on the fact that solutions of integral equations can be expressed in terms of solutions of some exterior and interior boundary value problems. In the author’s papers [3–4] asymptotics of solutions of boundary integral equations near corner points of the contour in plane problems of elasticity of the first two boundary value problems for the Lame’s system was obtained.

Stepan Zargaryan

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