The Dirichlet problem by variational methods

Wolfgang Arendt and Daniel Daners
Preprint February 2007
Bulletin of the London Mathematical Society, 40 (2008), 51 - 56.
Original article at doi:10.1112/blms/bdm091
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Abstract

Let ΩRN be an bounded open set and φC(Ω). Assume that φ has an extension ΦC(Ω¯) such that ΔΦH1(Ω). Then by the Riesz representation theorem there exists a unique uH01(Ω)such thatΔu=ΔΦin D(Ω). We show that u+Φ coincides with the Perron solution of the Dirichlet problem Δh=0,h|Ω=φ. 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.

AMS Subject Classification (2000): 35J05, 31B05

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