Hansen, W.; Netuka, I. Locally uniform approximation by solutions of the classical Dirichlet problem. (English) Zbl 0773.31010 Potential Anal. 2, No. 1, 67-71 (1993). Let \(U\) be an open subset of a \({\mathcal P}\)-harmonic space. It is shown that, in the topology of local uniform convergence, the space of classical solutions of the Dirichlet problem on \(U\) is dense in the space of Perron-Wiener-Brelot solutions of the Dirichlet problem if and only if the irregular boundary points of \(U\) form a set of harmonic measure 0. Reviewer: D.H.Armitage (Belfast) Cited in 1 ReviewCited in 2 Documents MSC: 31D05 Axiomatic potential theory 35A35 Theoretical approximation in context of PDEs Keywords:harmonic space; irregular boundary points PDF BibTeX XML Cite \textit{W. Hansen} and \textit{I. Netuka}, Potential Anal. 2, No. 1, 67--71 (1993; Zbl 0773.31010) Full Text: DOI References: [1] Bliedtner, J. and Hansen, W.:Potential Theory ? An Analytic and Probabilistic Approach to Balayage, Universitext, Springer-Verlag, Berlin (1986). · Zbl 0706.31001 [2] Brelot, M.: Remarque sur le prolongement fonctionnel et le problème de Dirichlet,Acta Sci. Math. (Szeged)12 (1950), 150-152. · Zbl 0036.06902 [3] Brelot, M.: Sur un théorème de prolongement fonctionnel de Keldych concernant le problème de Dirichlet,J. Anal. Math. 8 (1960/61), 273-288. · Zbl 0111.09604 [4] Constantinescu, C. and Cornea, A.:Potential Theory on Harmonic Spaces, Springer-Verlag, Berlin (1972). · Zbl 0248.31011 [5] Landkof, N.:Foundations of Modern Potential Theory, Springer-Verlag, Berlin (1971). · Zbl 0253.31001 [6] Netuka, I.: La représentation de la solution généralisée à l’aide des solutions classiques du problème de Dirichlet, in:Séminaire de Théorie du Potentiel, Lecture Notes in Mathematics906, Springer-Verlag, Berlin (1982), 261-268. [7] Netuka, I.: Approximation by harmonic functions and the Dirichlet problem, in:Approximation by Solutions of Partial Differential Equations (eds.: B. Fugledeet al.), Proceedings of NATO ASI Series, Ser. C, vol.365, Kluwer Academic Publishers, Dordrecht (1992), 155-168. · Zbl 0759.41033 [8] Schirmeier, H. and Schirmeier, U.: Einige Bemerkungen über den Satz von Keldych,Math. Ann. 236 (1978), 245-254. · Zbl 0378.31008 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.