Neumann Boundary Value Problem in a Half Plane

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Find all bounded solutions to the PDE $\Delta u(x,y) = 0$ for $x\in \mathbb{R}$ and $y > 0$ with Neumann boundary condition $u_y(x,0) = g(x)$.

 

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Spoiler

Let $\hat{u}(k,y) = \int_{-\infty}^\infty u(x,y)e^{ikx}\, dx$, the Fourier transform of $u$ with respect to the first variable. Applying this Fourier transform to the PDE with respect to the first variable yields $\hat{u}_{yy} - k^2 \hat{u} = 0$ with $\hat{u}_y(k,0) = \hat{g}(0)$. We have general solution $\hat{u}(k,y) = A(k)e^{ky} + B(k) e^{-ky}$. Assuming boundedness of $u$, $A(k) \equiv 0$ if $k > 0$ and $B(k) \equiv 0$ if $k < 0$. So we express $\hat{u}(k,y) = C(k)e^{-|k|y}$. The condition $\hat{u}_y(k,0) = \hat{g}(k)$ forces $-|k| C(k) = \hat{g}(k)$. Therefore $\hat{u}_y(k,y) = -|k| C(k) e^{-|k|y} = \hat{g}(k) e^{-|k|y}$. By the convolution theorem we obtain $$u_y(x,y) = \frac{y}{\pi} \int_{-\infty}^\infty \frac{g(x-t)}{t^2 + y^2}\, dt$$ Integrating with respect to $y$ produces solution $u(x,y) = \frac{1}{2\pi} \int_{-\infty}^\infty g(x-t) \log(t^2 + y^2)\, dt + c$.