**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 $0\in\mathbb{R}^2$. 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 $K\subset B$ be a compact set and $g:K\rightarrow \mathbb{R}$ 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 $\alpha\in\mathbb{R}$ such that it is satisfied


$$
\lim\_{x\longrightarrow z}\frac{|u\_s(x)-u\_s(z)|}{||x-z||^{\alpha}} = C(s, z) \neq 0,
$$
where $C(s, z)$ is a constant, depending on $s$ and $z\in K$?


Note that for $s=0$, the above limit exists when $\alpha = 1$ and $C(0)$ is the Lipschitz constant of of $u\_0$. 



Asymptotic behaviour near the boundary in the Dirichlet problem for the Laplacian.
No, it will not be differentiable in the whole ball. To see this, let $u$ be the zero function and $g$ be nearly anything nonnegative and not identically zero in $K$. For example $g=1$. Then recall Hopf's lemma.


This will also work to show that differentiability fails at any point on the boundary of $K$, at which $g$ achieves its maximum (on the whole of $K$).


However, it will be $C^\alpha$ in the ball, which is the last question you stated. This follows from the smoothness of $g$ and Holder estimates for $u$. For this you also need something about $K$ itself being smooth of course-- all hope is lost if the boundary of $K$ is irregular.


