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I read this sufficient condition for solvable boundary problem (Gilbarg & Trudinger 1998):

"the boundary value problem is solvable if every component of the complement of the domain consists of more than a single point."

Could anyone suggest where I may find a formal proof of this, or kindly provide me with a barrier function to show the statement holds? Thanks in advance!

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One place to check this out is here

You may also take a look at this recent paper by R. Green and K-T. Kim, The Riemann mapping theorem from Riemann's viewpoint. Start on page 6, where some discussion on your quest transpires.

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  • $\begingroup$ Yeah I read those two indeed. But there is no formal proof nor example of barrier functions. Thanks all the same tho! $\endgroup$
    – Sue Lu
    Commented May 26, 2017 at 3:32
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If they are talking about the Laplace operator, this statement is true only in dimension 2. And this is only sufficient, not necessary.

In general, for solvability of the classical Dirichlet problem, the boundary has to be regular (in the sense of potential theory). A necessary and sufficient condition of regularity is given by the Wiener criterion (see, for example Landkof, Introduction to modern potential theory, or any other book on potential theory). In dimension 2 the following is true: if a point is contained in a connected closed subset of the boundary, then it is regular. This is sufficient but not necessary. For example, the standard Cantor set is regular at each of its points. In dimension n, a smooth surface of codimension 2 consists of irregular points.

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  • $\begingroup$ Thanks for the reply! Yes true. Lebesgue provides an example for higher dimension, in a french book so I didn't look into it, not sure if there is proof for 2-d. $\endgroup$
    – Sue Lu
    Commented May 26, 2017 at 3:40

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