2006/08/10 by Wolfhard Hansen, Hansen, Wolfhard, Ivan Netuka +1
Computer Science · Mathematics · #30C85 #31A015 #31A05 #31B05 #31B15 #31C15 #46A55 #60G52 #60J65 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:30C85 #msc:31A015 #msc:31A05 #msc:31B05 #msc:31B15 #msc:31C15 #msc:46A55 #msc:60G52 #msc:60J65
paper · pdf · doi:10.48550/arxiv.math/0608268
arxiv created 2006/08/10 · openalex publication_date 2006/08/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is shown that, given a point x∈\mathbbmRd, d≥ 2, and open sets U1,...,Uk containing x, any convex combination of the harmonic measures for x with respect to Un, 1≤ n≤ k, is the limit of a sequence of harmonic measures for x with respect to open subsets Wm of U1∪... ∪ Uk containing x. This answers a question raised in connection with Jensen measures. More generally, we prove that, for arbitrary measures on an open set W, the set of extremal representing measures, with respect to the cone of continuous potentials on W or with respect to the cone of continuous functions on the closure of W which are superharmonic W, is dense in the compact convex set of all representing measures. This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then shrinking these balls in a suitable manner. The results are presented simultaneously for the classical case and for the theory of Riesz potentials. Finally, a characterization of all Jensen measures and of all extremal Jensen measures is given.