While reading Jost's PDE book, I came across a passage that I don't quite understand. The section discusses how to utilize a Green function to solve Laplace's Equation with Dirichlet Boundary conditions. Then, he says
"Analogously, if $H(x,y)$ for $x,y\in \bar{\Omega}$, $x\neq y$ is defined with $$\frac{\partial}{\partial \nu_x} H(x,y) = \frac{1}{||\partial\Omega||} \text{ for } x\in \partial\Omega$$ (Here, $||\partial\Omega||$ denotes the measure of the boundary $\partial\Omega$ of $\Omega$; it is given as $\int_{\partial\Omega} do(x)$) and a harmonic difference $H(x,y)-\Gamma(x,y)$ as before, we obtain $$u(y) = \frac{1}{||\partial\Omega||}\int_{\partial\Omega} u(x) do(x) - \int_{\partial\Omega} H(x,y)\frac{\partial u}{\partial \nu}(x) do(x) + \int_\Omega H(x,y)\Delta u(x) dx."$$ But, I don't see how this would be helpful in solving the Neumann Boundary condition: $$\Delta u = 0 \text{ on } \Omega$$ $$\frac{\partial u}{\partial \nu} = g \text{ on } \partial\Omega.$$ I would have thought that one would wish to find a function $H(x,y)$ as above, but instead of requiring $$\frac{\partial}{\partial \nu_x} H(x,y) = \frac{1}{||\partial\Omega||} \text{ for } x\in \partial\Omega,$$ we ask that $$\frac{\partial}{\partial \nu_x} H(x,y) = 0 \text{ for } x\in \partial\Omega.$$ This way, the solution to the Neumann Boundary problem would simply be $$u(y) = -\int_{\partial \Omega} H(x,y) \frac{\partial u}{\partial \nu}(x) do(x).$$ Why does Jost bring up this function instead? Is there something I am missing?