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Consider the PDE $Lu = 0$ in the upper half plane, where $L$ is a second order elliptic differential operator. On $y=0$, we give it the boundary condition $a u + b \frac{\partial u}{\partial \nu} = 0$, where $\nu$ is the inwards pointing unit normal vector.

My question is, is there a standard way of extending $u$ in a tubular neighborhood of the $x$-axis, that is, converting the problem to a PDE without boundary? If the boundary condition were Dirichlet/Neumann, we could do so by taking the odd/even extension in the $y$-variable, for example. But I am unsure about the general Robin conditions.

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  • $\begingroup$ A reference for "Robin conditions" ? $\endgroup$ – Jean Marie Jan 16 '17 at 22:32
  • $\begingroup$ @JeanMarie Looked at a few, but not finding anything reasonable. I would be very grateful if you could point to one. $\endgroup$ – user346998 Jan 17 '17 at 12:01
  • $\begingroup$ It is doable but it is more challenging than homogeneous Dirichlet and Neuman Boundary Conditions. You can follow section 2.2.3 of this document: studocu.com/en/document/technische-universiteit-delft/… $\endgroup$ – pluton Mar 14 at 11:49

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