×

zbMATH — the first resource for mathematics

Harmonic continuation and removable singularities in the axiomatic potential theory. (English) Zbl 0358.31007
Let \(U\) be a relatively compact open subset of a harmonic space \(X\) and \(S\) be the set of all functions continuous on \(\bar{U}\) and superharmonic on \(U\). The Šilov boundary of \(U\) with respect to \(S\) will be denoted by \(\partial_S\bar{U}\) and \(H = S \cap (-S)\) will be considered as a Banach space (\(H\) is equipped with the supremum norm). Suppose that \(x \in \partial U\) and \(h \in H\). Then \(x\) is termed a point of continuability of \(h\), if there is a function \(h_1\) harmonic on a neighborhood \(V\) of \(x\) such that \(h=h_1\) on \(V \cap U\). Assuming that the points of \(\partial U\) are polar, the typical result concerning the possibility of harmonic continuation reads as follows: Each point of \(\partial U\setminus \partial_S \bar{U}\) is a point of continuability of any \(h\in H\) and the set of all functions of \(H\), for which no point of \(\partial_s \bar{U}\) is a point of continuability, is a dense \(G_{\delta}\) in \(H\). Suppose that \(F\subset X\) is a closed set. We say that \(F\) has \(c\)-capacity zero provided there is no non-trivial potential with support in \(F\). Further \(\beta(F)\) is the essential base of \(F\). The following theorem on removable singularities for continuous (super) harmonic functions is proved. Conditions (i) - (v) are equivalent:
(i) F is semi-polar.
(ii) (respectively (iii)) If \(G\) is an open set and \(f\) is a continuous function on \(G\) and superharmonic (respectively harmonic) on \(G\setminus F\), then \(f\) is superharmonic (respectively harmonic) on \(G\).
(iv) \(F\) has \(c\)-capacity zero.
(v) \(\beta(F)=\emptyset\).
This theorem represents a generalization of some results of J. Köhn and M. Sieveking [Revue Roumaine Math. pure. appl. 12, 1489-1502 (1967; Zbl 0158.12804)] and R. Harvey and J. C. Polking [Trans. Amer. math. Soc. 169, 183-195 (1972; Zbl 0249.35012)]. Proofs of the theorems mentioned above use the theory of simplicial cones developed by J. Bliedtner and W. Hansen [Inventiones Math. 29, 83-110 (1975; Zbl 0308.31011].

MSC:
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31C15 Potentials and capacities on other spaces
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Anger, G.: Funktionalanalytische Betrachtungen bei Differentialgleichungen unter Verwendung von Methoden der Potentialtheorie. I. Berlin: Akademie-Verlag 1967 · Zbl 0163.11901
[2] Bauer, H.: ?ilovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier11, 89-136 (1961) · Zbl 0098.06902
[3] Bauer, H.: Harmonische R?ume und ihre Potentialtheorie. Lecture Notes in Mathematics 22. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0142.38402
[4] Bliedtner, J., Hansen, W.: Simplicial cones in potential theory. Invent. Math.29, 83-110 (1975) · Zbl 0308.31011
[5] Constantinescu, C., Cornea, A.: Potential theory on harmonic spaces. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0248.31011
[6] Harvey, R., Polking, J. C.: A notion of capacity which characterizes removable singularities. Trans. Amer. Math. Soc.169, 183-195 (1972) · Zbl 0249.35012
[7] K?hn, J., Sieveking, M.: Regul?re und extremale Randpunkte in der Potentialtheorie. Rev. Roum. Math. Pures Appl.12, 1489-1502 (1967) · Zbl 0158.12804
[8] Kr?l, J., Luke?, J.: Indefinite harmonic continuation. ?asopis P?st. Mat.98, 87-94 (1973)
[9] Luke?, J., Netuka, I.: The Wiener type solution of the Dirichlet problem in potential theory. Math. Ann.224, 173-178 (1976) · Zbl 0327.31009
[10] Pradelle, A. de la: Approximation et caract?re de quasi-analyticit? dans la th?orie axiomatique des fonctions harmoniques. Ann. Inst. Fourier17, 383-399 (1967) · Zbl 0153.15501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.