Potential Theory on Compact Sets

The primary goal of this work is to extend the notions of potential theory to compact sets. There are several equivalent ways to define continuous harmonic functions H(K) on a compact set K in Rn. One may let H(K) be the uniformclosure of all functions in C(K) which are restrictions of harmonic functions on a neighborhood of K, or take H(K) as the subspace of C(K) consisting of functions which are finely harmonic on the fine interior of K. In [9] it was shown that these definitions are equivalent.;We study the Dirichlet problem on a compact set K in Chapter 4. As in the classical theory, our Theorem 4.1 shows set of continuous functions on the fine boundary is isometrically isomorphic to H(K) for compact sets whenever the fine boundary is closed in the Euclidean topology. However, in general a continuous solution cannot be expected even for continuous data on the fine boundary. Consequently, Theorem 4.3 shows that the solution can be found in a class of finely harmonic functions.;To study these spaces, two notions of Green functions have previously been introduced. One by [22] as the limit of Green functions on a sequence of domains decreasing to K. Alternatively, following [12, 13] one has the fine Green function on the fine interior of K. Our Theorem 3.14 shows that these are equivalent notions.;Using a localization result of [3] one sees that a function is in H(K) if and only if it is continuous and finely harmonic on every fine connected component of the fine interior of K. Such collection of sets is usually called a restoring covering. Another equivalent definition of H(K) was introduced in [22] using the notion of Jensen measures which leads to another restoring collection of sets.;In Section 5.1 a careful study of the set of Jensen measures on K, leads to an interesting extension result (Corollary 5.8) for subharmonic functions. This has a number of applications. In particular we show that the restoring coverings of [9] and [22] are the same. We are also able to extend some results of [18] and [22] to higher dimensions.