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Bulletin of the London Mathematical Society 2008 40(1):51-56; doi:10.1112/blms/bdm091
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© 2008 London Mathematical Society

The Dirichlet problem by variational methods

Wolfgang Arendt

Institute of Applied Analysis
University of Ulm
D–89069 Ulm
Germany

Daniel Daners

School of Mathematics and Statistics
The University of Sydney
NSW 2006
Australia
D.Daners@maths.usyd.edu.au

Received 15 March 2007. Revision received 13 July 2007.

Let {Omega}subRN be a bounded open set and {varphi}isinC({partial} {Omega}). Assume that {varphi} has an extension {Phi}isinC(Formula) such that {Delta} {Phi}isinH–1({Omega}). Then by the Riesz representation theorem there exists a unique


Formula

We show that u+{Phi} coincides with the Perron solution of the Dirichlet problem


Formula

This extends recent results by Hildebrandt [Math. Nachr. 278 (2005), 141–144] and Simader [Math. Nachr. 279 (2006), 415–430], and also gives a possible answer to Hadamard's objection against Dirichlet's principle.

2000 Mathematics Subject Classification 35J05, 31B05.


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