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Über das Verhalten der Lösung des Dirichletproblems am Rand des Gebietes, wenn der Rand zur Klasse C2α gehört

Published online by Cambridge University Press:  14 November 2011

Manfred König
Manfred König
Affiliation:
Edelweiss Strasse 11, 8131 Berg-4-Höhenrain, Germany
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Synopsis

In this paper the boundary behaviour of the solution of Dirichlet's problem for the equation Δu = f in domains with C2,α-boundary is studied. Without using Schauder's a priori estimates, we prove the following theorem, which is indispensable in Schauder's method to solve Dirichlet's problem for a general linear elliptic equation.

Theorem: Let (f, g)∈ C0,α (Ḡ)×C2,α (∂G) and u ∈ C0.0(Ḡ) ∩C2,α(G) be a solution of the problem Δu = f, u|∂G = g. Then u ∈ C2,α(Ḡ).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 1-2 , 1978 , pp. 163 - 176
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Barrar, R. B.On Schauder's Paper on Linear Elliptic Differential Equations. J. Math. Anal. Appl. 3 (1961), 171–195.CrossRefGoogle Scholar
2Boboc, N. and Mustatǎ, P.Remarks on the existence of solutions of the Dirichlet problem for strongly elliptic linear operators of second order. Bull. Math. Soc. Sci. Math. R.S. Roumanie 10 (1966), 75–85.Google Scholar
3Graves, L. M.The Estimates of Schauder and their Applications to Existence Theorems for Elliptic Differential Equations. Chicago Univ. Invest. Theory Partial Differential Equations Techn. Report 1 (1956).Google Scholar
4Günter, N. M.Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der mathematischen Physik (Leipzig: Teubner, 1957).Google Scholar
5Hellwig, G.Partielle Differentialgleichungen (Stuttgart: Teubner, 1960).Google Scholar
6Hopf, E.Über den funktionalen, insbesondere den analytischen Charakter der Lösungen elliptischer Differentialgleichungen zweiter Ordnung. Math. Z. 34 (1932), 194–233.CrossRefGoogle Scholar
7Kellogg, O. D.On the derivatives of harmonic functions on the boundary. Trans. Amer. Math. Soc. 33 (1931), 486–510.CrossRefGoogle Scholar
8Ladyzhenskaya, O. A. and Uralt'seva, N.N.Linear and Quasilinear Elliptic Equations (London: Academic Press, 1968).Google Scholar
9Leis, R.Vorlesungen über partielle Differentialgleichungen zweiter Ordnung (Mannheim: B. I. Hochschultaschenbücher, 1967).Google Scholar
10Müntz, Ch.Zum Randwertproblem der partiellen Differentialgleichungen der Minimalfläche. J. Reine Angew. Math. 139 (1911), 52–79.CrossRefGoogle Scholar
11Perron, O.Eine neue Behandlung der ersten Randwertaufgabe für Δu = 0. Math. Z. 18 (1923), 42–54.CrossRefGoogle Scholar
12Schauder, J.Potentialtheoretische Untersuchungen. Math. Z. 33 (1931), 602–640.CrossRefGoogle Scholar
13Schauder, J.Über lineare elliptische Differentialgleichungen zweiter Ordnung. Math. Z. 38 (1934), 257–282.CrossRefGoogle Scholar
14Simoda, P. S.Sur le théorème de Müntz dans la théorie du potential. Osaka J. Math. 3 (1951), 65–75.Google Scholar
15Widman, K. O.Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand. 21 (1967), 17–37.CrossRefGoogle Scholar