Let $L=\{ (x,y): x^2+y^2<5\}$.
Suppose that we have a function $u(x,y)$ that is harmonic in $L$ and $\Omega \subset L$ an arbitrary space such that $(0,0)$ belongs to that space.
Suppose that $u$ is equal to $1$ at the boundary of $\Omega$.
How can we compute the integral $\int_{x^2+y^2 \leq 4} u(x,y) dxdy$ ?
I thought that we could use the second version of the mean value theorem.
The theorem is the following:
If $u$ is harmonic in $\Omega$ and $\overline{x} \in \overline{\Omega} \setminus \partial{\Omega}$ then $u(\overline{x})=\frac{n}{ w_n r^n} \int_{|\overline{x}-\overline{\xi}| \leq …