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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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closed as too broad by Michael Renardy, Yemon Choi, David Roberts, Asaf Karagila, Carlo Beenakker Jun 28 '13 at 9:53

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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 $$ u(x,y,z)=-\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}}=\frac{g(\mu,\nu)}{\sqrt{(x-\mu)^2+(y-\nu)^2+z^2}}d\nu d\mu. $$

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