I want to slove the Laplace equation on $R^3_+$ with Neumann boundary condition. The equation reads: $-\Delta u = f$ in $R^3_+$, $\partial_3 u|_{x_3=0}=g$ on $R^2$. If $f$, $g$ satisfy compatibility condition, can I write the explicit formular of $u$?
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Maybe the book "Handbook of Linear Partial Differential Equations for Engineers and Scientists" by Polyanin, A. D. Chapman & Hall/CRC, 2002, can help you.
By the way, if $f=0$ the solution (up to a constant because the solution is not unique) is $$ u(x,y,z)=-\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}}=\frac{g(\mu,\nu)}{\sqrt{(x-\mu)^2+(y-\nu)^2+z^2}}d\nu d\mu. $$
