Equivalence of weak Dirichlet's principle, the method of weak solutions and Perron's method towards classical solutions of Dirichlet's problem for harmonic functions

For boundary data ϕ ∈ W 1,2(G ) (where G ⊂ ℝN is a bounded domain) it is an easy exercise to prove the existence of weak L 2‐solutions to the Dirichlet problem “Δu = 0 in G, u |∂G = ϕ |∂G ”. By means of Weyl's Lemma it is readily seen that there is ũ ∈ C ∞(G ), Δũ = 0 and ũ = u a.e. in G . On the contrary it seems to be a complicated task even for this simple equation to prove continuity of ũ up to the boundary in a suitable domain if ϕ ∈ W 1,2(G ) ∩ C 0($ {\overline {G} } $). The purpose of this paper is to present an elementary proof of that fact in (classical) Dirichlet domains. Here the method of weak solutions (resp. Dirichlet's principle) is equivalent to the classical approaches (Poincaré's “sweeping‐out method”, Perron's method). (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)