I'm kind of stuck on the following exercise:
Suppose $u \in C^2(\Omega)$. Given $x \in \Omega$, show that
\begin{align} \Delta u(x) = \lim_{r \to 0} \frac{2n}{r^2} \left[ \frac{1}{w_n} \int_{\partial B_{1}} u(x + ry)\mathrm{d}S_y - u(x) \right]. \end{align}
Conclude that from the above reciprocal expression of Mean Value Property.
We have a suggestion to solve it:
Consider Taylor's second-order polynomial of $u$ around $x$. Using symmetry, show that
\begin{align} \int_{\partial B_{1}} x_j \mathrm{d}S = \int_{\partial B_{1}} x_j x_k \mathrm{d}S = 0, \,\,\,\mathrm{para} \,\,\,j\neq k \end{align} and \begin{align} \int_{\partial B_{1}} x_k^{2} \mathrm{d}S = n ^{-1}\int_{\partial B_{1}} \left[ \sum_{j=1}^{n} x_j^{2}\right] \mathrm{d}S = \int_{\partial B_{1}} 1 \mathrm{d}S. \end{align}
Can anyone give me some help with this exercise?
Thanks in Advance.