ap.analysis of pdes - Asymptotic behaviour near the boundary in the Dirichlet problem for the Laplacian.

Perturbative behaviour of solutions of the solutions of the Dirichlet problem for the Laplacian:

Lets consider $ B = B(0, 1) in mathbb{R}^2$ be the unit circle with center at $0inmathbb{R}^2$. Let $u_0$ be an harmonic function on $B$ also harmonic at the boundary, that is, $u_0$ is harmonic in the ball $B(0, 1+varepsilon)$ for $varepsilon > 0$ small. Then, if we denote by $f = {u_0}_{|partial B}$ we have that $u_0$ satisies (trivially) the Dirichlet problem

$$
begin{array} {rcl}
Delta u_0(x) & = & 0 newline
{u_0}_{|partial B}(x) &= &f(x)
end{array}
$$

Now, let $Ksubset B$ be a compact set and $g:Krightarrow mathbb{R}$ be a smooth function (real analytic, for instance), and consider the one parameter family of Dirichlet problems

$$
begin{array} {rcl}
Delta u_s(x) & = & 0 newline
{u_s}_{|partial B}(x) &= &f(x)newline
{u_s}_{|K}(x) &= & {u_0}_{|K}(x)+sg(x)newline
end{array}
$$

It is clear that for $s=0$ the solution of this problem is the same as the original problem stated above, so we consider this as a perturbative problem.

MY QUESTION IS:

How does $u_s$ behaves near the compact set $K$? It is known that $u_s$ is continuous in all the unit ball (also in $K$) but it is hoped that is not differentiable near $K$. It is possible to show that, generically, there exists an $alphainmathbb{R}$ such that it is satisfied

$$
lim_{xlongrightarrow z}frac{|u_s(x)-u_s(z)|}{||x-z||^{alpha}} = C(s, z) neq 0,
$$
where $C(s, z)$ is a constant, depending on $s$ and $zin K$?

Note that for $s=0$, the above limit exists when $alpha = 1$ and $C(0)$ is the Lipschitz constant of of $u_0$.