TITLE: Dirichlet Problem with piecewise smooth boundary
QUESTION [5 upvotes]: Suppose a domain $ \Omega \subset \mathbb{R^2} $ with $ \partial \Omega $. For $ f \in C^{\infty}(\mathbb{R^2}) $, the dirichlet problem is to find $ u $ with $ \Delta u = 0 $ in $ \Omega $, and $ f = u $ on $ \partial \Omega $. What are the existence and regularity theorems for classical solutions $ u $ in the case that $ \partial \Omega $ is not smooth, but only piecewise smooth?
I am not sure exactly where to look for these results beyond just knowing the standard references, and I am a little overwhelmed by this. I would grateful if someone could put point me in a better direction.

REPLY [2 votes]: Well, if you take the Perron method, you always have existence (take, for example, the pointwise supremum of all subharmonic functions that lie below $f$ on $\partial \Omega$), and for harmonic functions you always have interior regularity (see, for example, the interior Schauder estimates in Gilbarg-Trudinger, Chapter 4 or Chapter 6). 
When the boundary is no longer smooth, what you may have are convergence issues - does your solution converge to the boundary value? If so, how quickly? That is settled by whether there is a barrier about that boundary point (Gilbarg-Trudinger also has a discussion of what this means more precisely in Chapter 2, Section 8 - "The Method of Subharmonic functions"), which relates to the geometry of that boundary point - this is quantified by the Wiener criterion. 
(Update): The following paper may be useful: http://link.springer.com/content/pdf/10.1007%2FBF00281357.pdf