The Riesz decomposition of finely superharmonic functions

This paper answers a question of Fuglede about minimal positive harmonic functions associated with irregular boundary points. As a consequence, an old and central problem of fine potential theory, concerning the Riesz decomposition, is resolved. Namely, it is shown that, on certain fine domains, there exist positive finely superharmonic functions which do not admit any positive finely harmonic minorant and yet are not fine potentials.     

Quadrature Domains with Infinite Volume

We discuss quadrature domains for subharmonic functions and prove the existence of core quadrature domains for certain positive measures. The core quadrature domains are the smallest quadrature domains as measures and inherit good properties from quadrature domains with finite volume. We next discuss new balayage for the class of harmonic functions integrable in a neighborhood of ∞. We give several estimates of balayage measures. The new balayage is introduced to construct quadrature domains for harmonic functions. Submitted: June 26, 2008. Accepted: July 24, 2008.     

Asymptotic Values and Minimal Fine Limits of Subharmonic Functions of Slow Growth

This paper shows that a subharmonic function in the half-spacewhich does not grow too rapidly near the boundary and whichdoes not have asymptotic value + at too many points must havefinite minimal fine limits at a boundary set of positive measure.For harmonic functions, the conclusion may be expressed in termsof non-tangential limits. A related Phragmén–Lindelöftheorem is also established.     

Complete submanifolds in manifolds of partially non-negative curvature

We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the same kind of submanifolds are also obtained.     

Approximation in the Cone of Positive Harmonic Functions

In the course of studying quadrature domains Gustafsson, Sakai and Shapiro were led to the question of whether it is the case that the positive integrable harmonic functions on a bounded domain are dense among all positive harmonic functions (w.r.t. uniform convergence on compact subsets). In this article we will show how such approximation problems are related to representing measures on the Martin boundary, and then we use these results to give a counterexample to the question posed above. This research was supported by Science Foundation Ireland under Grant 06/RFP/MAT057.     

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

We define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

Harmonic manifolds with minimal horospheres

For a non-compact harmonic manifold M, we establish an integral formula for the derivative of a harmonic function on M. As an application we show that for the harmonic spaces having minimal horospheres, bounded harmonic functions are constant. The main result of this article states that the harmonic spaces having polynomial volume growth are flat. In other words, if the volume density function Θ of M has polynomial growth, then M is flat. This partially answers a question of Szabo namely, which density functions determine the metric of a harmonic manifold. Finally, we give some natural conditions which ensure polynomial growth of the volume function.     

Asymptotic of the heat kernel in general Benedicks domains

 Using a new inequality relating the heat kernel and the probability of survival, we prove asymptotic ratio limit theorems for the heat kernel (and survival probability) in general Benedicks domains. In particular, the dimension of the cone of positive harmonic measures with Dirichlet boundary conditions can be derived from the rate of convergence to zero of the heat kernel (or the survival probability). Received: 31 March 2002 / Revised version: 12 August 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 60J65, 31B05 Key words or phrases: Positive harmonic functions – Ratio limit theorems – Survival probability     

Beurling's minimum principle in a cylinder

This article gives two types of conditions for a set E in a cylinder such that all positive harmonic functions in the cylinder which (essentially) majorize a minimal function at +∞ on E majorize it on the whole cylinder.     

Applications of nonstandard analysis to ideal boundaries in potential theory

A solution is given of the generalized Dirichlet problem for an arbitrary compactification of a Brelot harmonic space. A method of obtaining the Martin-Choquet integral representation of positive harmonic functions is given, and the existence is established of an ideal boundary Δ supporting the maximal representing measures for positive bounded and quasibounded harmonic functions with almost all points of Δ being regular for the Dirichlet problem. This work was supported by a grant from the U. S. National Science Foundation. The results in Sections 1–5 were presented at the 1974 Oberwolfach Conferences on Potential Theory and Nonstandard Analysis; Sections 1–6 were discussed at the Abraham Robinson Memorial Conference, Yale, University, May 1975.     

Periodic harmonic functions on lattices and points count in positive characteristic

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.      

Hua operator on vector bundle: Application to AdS/CFT correspondence of Dirac fields

Hua’s theory of harmonic functions on classical domains is generalized to the theory on holomorphic vector bundles over classical domains and further on vector bundles over the real classical domains and quaternion classical domains. In case of the simplest quaternion classical domain there is a relation between Hua operator and Dirac operator, by which an AdS/CFT correspondence of Dirac fields is established.     

Potential Theory of Special Subordinators and Subordinate Killed Stable Processes

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D. The research of this author is supported in part by MZOS grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

有界多连通区域数值保角变换的GMRES(m)法北大核心CSCD

求解复杂多连通区域的保角变换函数是困难的.针对这一问题,该文将求解保角变换函数转化为利用模拟电荷法求解一对定义在问题区域上的共轭调和函数,再根据边界条件建立约束方程,并利用GMRES(m)(the generalized minimal residual method)算法求解约束方程,获得了模拟电荷,进而构造了高精度的近似保角变换函数,将有界多连通区域映射为三种无界正则狭缝域.数值实验验证了该文算法的有效性.     

Positive Harmonic Functions for Semi-Isotropic Random Walks on Trees, Lamplighter Groups, and DL-Graphs

We determine all positive harmonic functions for a large class of “semi-isotropic” random walks on the lamplighter group, i.e., the wreath product ℤ_q≀ℤ, where q≥2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel–Leader graph . More generally, (q,r≥2) is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1, and our result applies to all -graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees. Mathematics Subject Classifications (2000) 60J50, 05C25, 20E22, 31C05, 60G50. Supported by European Commission, Marie Curie Fellowship HPMF-CT-2002-02137.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Propagation of Smallness for Harmonic and Analytic Functions in Arbitrary Domains

In this paper the authors develop a new approach to the problemof ‘propagation of smallness’ for harmonic functionsin arbitrary domains, in Rⁿ (n 2). The main result of thispaper is a certain logarithmic-convexity relation for the L²-normsof harmonic functions. As a consequence, new kinds of uniquenessresults for harmonic functions are obtained. The method worksalso for analytic functions in C, with L^p-norms (p > 0).1991 Mathematics Subject Classification 31B05.     

Martin Kernels for Markov Processes with Jumps

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R ^d with positive continuous density of the Lévy measure; stable-like processes in R ^d and in domains; and stable-like subordinate diffusions in metric measure spaces.     

Extended Harnack inequalities with exceptional sets and a boundary Harnack principle

Harnack’s inequality is one of the most fundamental inequalities for positive harmonic functions and has been extended to positive solutions of general elliptic equations and parabolic equations. This article gives a different generalization; namely, we generalize Harnack chains rather than equations. More precisely, we allow a small exceptional set and yet obtain a similar Harnack inequality. The size of an exceptional set is measured by capacity. The results are new even for classical harmonic functions. Our extended Harnack inequality includes information about the boundary behavior of positive harmonic functions. It yields a boundary Harnack principle for a very nasty domain whose boundary is given locally by the graph of a function with modulus of continuity worse than Hölder continuity.     

Horizontal Growth of Harmonic Functions

We show that positive harmonic functions in the upper halfplane grow at most quadratically in horizontal bands. This bound is sharp in a sense to be specified, which, at least implies that there are examples growing as fast as any power under 2. These results are extended to positive harmonic functions in a half-space of R ^{n +1}, with points represented by ( x , y ), where x ∈R ⁿ , and y ∈R, the sharp maximum rate of growth being now ¦ x ¦ ^{n +1}. The case of Poisson integrals of functions in L_p ( dx /(1+(¦ x ¦)² )^{( n +1)/2}) is also taken up; the bound condition is then O (¦ x ¦^{( n +1)/ p} ).