I have the following question that starts from this expression: $$ u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y) . $$
If $h \in C^2(\bar{B})$ is harmonic and Green function $G(x, y)=\Gamma(y-x)-h(y)$, the above integral leads us to $$ u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) G(x, y)+u(y) \frac{\partial G}{\partial n_y}(x, y)\right) d \sigma(y) . $$
I have tried to develop it but I have not been able to reach the second integral. The derivative that appears is the normal derivative used in PDEs and its expression is $\frac{\partial u}{\partial n}=(\nabla u)^{T}\cdot n$ where $n$ is a vector. Developing the integrand, I have reached this point: \begin{equation*} -\frac{\partial u(y)}{\partial n}\cdot ( G(x,y) + h(y) ) + u(y)\cdot\frac{\partial (G(x,y) + h(y))}{\partial n} \end{equation*} \begin{equation*} -\frac{\partial u(y)}{\partial y}\cdot ( G(x,y) + h(y) ) + u(y)\cdot \nabla (G(x,y)+h(y))^{T}\cdot n \end{equation*} \begin{equation*} -\frac{\partial u(y)}{\partial y}\cdot ( G(x,y) + h(y) ) + u(y)\cdot \left(\frac{\partial ( G + h)}{\partial x} , \frac{\partial (G +h)}{\partial y}\right) \cdot (n_x , n_y) \end{equation*} \begin{equation*} -\frac{\partial u(y)}{\partial y}\cdot ( G(x,y) + h(y) ) + u(y)\cdot \left(\frac{\partial ( G )}{\partial x} , \frac{\partial (G +h)}{\partial y}\right) \cdot (n_x , n_y) \end{equation*} If we continue operating and simplifying, it can be seen that we do not reach the desired expression. Is the procedure wrong or is there a calculation error?
This whole problem comes from the book "Stable solutions of elliptic partial differential equations" by Louis Dupaigne, page 235.
given any function $g \in C(\partial B)$, there exists a unique function $u \in C^2(B) \cap C(\bar{B})$ solving $$ \left\{\begin{array}{rll} -\Delta u=0 & \text { in } B, & \\ u=g & \text { on } \partial B . & \end{array}\right. $$ Suppose for the moment that $u \in C^2(\bar{B})$ is a solution of this problem. Fix a point $x \in B$, multiply by $\Gamma(y-x)$, where $\Gamma$ is the fundamental solution of the Laplace operator and integrate over $B \backslash B(x, \varepsilon)$, $\varepsilon>0$. Then, $$ \begin{array}{cc} 0=\int_{B \backslash B(x, \varepsilon)}-\Delta u(y) \Gamma(y-x) d y= & \\ \int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y)- & \begin{array}{l} \end{array} \\ \int_{\partial B(x, \varepsilon)}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y) . \end{array} $$
Letting $\varepsilon \rightarrow 0$, it is not hard to deduce that cuando integras va a tender a $$ u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \sigma(y) . $$ And this continue above in my intial answer.
And the expression of $\Gamma$ when $N\neq 2$ is
\begin{equation}
\Gamma(x)=\Gamma(r)=\frac{1}{N(N-2)|B|}r^{2-N}
\end{equation}
where $|B|$ denotes the volume of the unit ball in $\mathbb{R}^N$, and by
\begin{equation}
\Gamma(x)=\Gamma(r)=-\frac{1}{2\pi}ln(r)
\end{equation}
if N=2- And where r=|x|, r^2=x^2_1 + \ldots + x^2_N