So, if $K$ is smooth, you state that $\alpha=1/2$. This is because there are examples in the unit ball with $K$ an interval on the real axis where $u$ is $C^{1/2}$ on the adherence of the ball.

@Scott Armstrong: I have two questions: 1.- Does the value $\alpha$ depends on $K$ and $g$? 2.- What happens if $K$ is a Cantor set? and $g$ is a smooth function on the ball $B$ restricted to $K$? Can we recover the $C^\alpha$ smoothness?

@ Willie Wong: I corrected the typo about the dependence of $C$ on $z$. About the literature, can you tell me exactly which are Gilbarg and Trudinger?

@Andrey Rekalo: The condition ${u_s}_{|K}(x) = {u_0}_{|K}(x)+sg(x)$ is not redundant. In fact, you can think that $K$ is part of the boundary for a new $\bar{\Omega} = \Omega-K$.

The solution must be harmonic in $B-K$ for all $s$.

I see.. The problem that I found is that it not assures that $p$ is in the intersection. Nevertheless, as you said, the Baire's theorem asserts that this intersection is dense but, by Borel-Cantelli lemma, that its measure is $0$.

Extremely good reference. Thanks for the article!!! I will enjoy a lot it!