where can I get a quick exposition to the boundary regularity problem for the Laplacian operator ?
In other words, suppose $h:S^1 \to \mathbb C $ and let $H: \bar{D}\to \mathbb C $ be its complex harmonic extension, i.e. $H(z) = \int h(t)p(z,t)\mathrm dt \quad \forall z\in D $ be the complex harmonic (but NOT holomorphic) extension of $h$ , where the integral is taken over $S^1$ and $p(z,t)$= Poisson kernel.
I want to quickly study the proof of the theorem : if the boundary data $h$ is $C^k$ , then the extension $H$ is $C^r(\bar{D})$ for some r .
I was told that the PDE book by Gilbarg-Trudinger is a source, but is there a quicker source where I can read everything in a short time ? Again, I just need in two dimensions.
Thanks !