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.