535 research outputs found
Martin boundary of a fine domain and a Fatou-Naim-Doob theorem for finely superharmonic functions
We construct the Martin compactification of a fine domain in
, , and the Riesz-Martin kernel on . We
obtain the integral representation of finely superharmonic fonctions on
in terms of and establish the Fatou-Naim-Doob theorem in this setting.Comment: Manuscript as accepted by publisher. To appear in Potential Analysi
Sweeping at the Martin boundary of a fine domain
We study sweeping on a subset of the Riesz-Martin space of a fine domain in
\RR^n (), both with respect to the natural topology and the
minimal-fine topology, and show that the two notions of sweeping are identical.Comment: Minor correctio
Domains of existence for finely holomorphic functions
We show that fine domains in with the property that they are
Euclidean and , are in fact fine domains of existence for
finely holomorphic functions. Moreover \emph{regular} fine domains are also
fine domains of existence. Next we show that fine domains such as
or , more specifically fine domains with the properties that
their complement contains a non-empty polar set that is of the first Baire
category in its Euclidean closure and that , are
NOT fine domains of existence.Comment: 13 pages 1 figure. This new version has Bent Fuglede as coauthor. We
extended the main result to include that regular fine domains are fine
domains of existence and corrected many typo's and inaccuracies. In the third
version a mistake at the end of the proof of Proposition 2.6 has been
correcte
Plurisubharmonic and holomorphic functions relative to the plurifine topology
A weak and a strong concept of plurifinely plurisubharmonic and plurifinely
holomorphic functions are introduced. Strong will imply weak. The weak concept
is studied further. A function f is weakly plurifinely plurisubharmonic if and
only if f o h is finely subharmonic for all complex affine-linear maps h. As a
consequence, the regularization in the plurifine topology of a pointwise
supremum of such functions is weakly plurifinely plurisubharmonic, and it
differs from the pointwise supremum at most on a pluripolar set. Weak plurifine
plurisubharmonicity and weak plurifine holomorphy are preserved under
composition with weakly plurifinely holomorphic maps.Comment: 28 page
The Dirichlet Problem for Harmonic Functions on Compact Sets
For any compact set we develop the theory of Jensen
measures and subharmonic peak points, which form the set , to
study the Dirichlet problem on . Initially we consider the space of
functions on which can be uniformly approximated by functions harmonic in a
neighborhood of as possible solutions. As in the classical theory, our
Theorem 8.1 shows for compact sets with
closed. However, in general a continuous solution cannot be
expected even for continuous data on \rO_K as illustrated by Theorem 8.1.
Consequently, we show that the solution can be found in a class of finely
harmonic functions. Moreover by Theorem 8.7, in complete analogy with the
classical situation, this class is isometrically isomorphic to
for all compact sets .Comment: There have been a large number of changes made from the first
version. They mostly consists of shortening the article and supplying
additional reference
Minimum Riesz energy problems for a condenser with "touching plates"
Minimum Riesz energy problems in the presence of an external field are
analyzed for a condenser with touching plates. We obtain sufficient and/or
necessary conditions for the solvability of these problems in both the
unconstrained and the constrained settings, investigate the properties of
minimizers, and prove their uniqueness. Furthermore, characterization theorems
in terms of variational inequalities for the weighted potentials are
established. The results obtained are illustrated by several examples.Comment: 32 pages, 1 figur
Spinor algebra and null solutions of the wave equation
In this paper we exploit the ideas and formalisms of twistor theory, to show
how, on Minkowski space, given a null solution of the wave equation, there are
precisely two null directions in , at least one of which is a
shear-free ray congruence
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