Is this PDE (Poisson/Laplace equation) well-posed considering I have a very degenerate domain (picture included)?

In case (1) in the following picture we have the standard interior Poisson equation in 2D with Neumann boundary conditions on some smooth domain $$\Omega$$, subject to a point source at position $$y$$.

I do not have much experience dealing with PDE's with singular geometry, I usually deal with smooth or Lipschitz boundaries so I am unsure of whether it is possible to consider the Poission equation in case (2) in which the domain/boundary extends as a line to both sides of the 'main' section of the domain. Its as if we squeezed the left and right sides of the domain in case (1) until they became infinitely thin leaving us with a circular central domain from which two lines extend.

I want to know whether it is possible to consider the Poisson equation in this situation or is it a completely invalid formulation of the PDE? The boundary is all that exists on the left and right lines, it terminates at the endpoints, and it doesn't enclose anything in these sections..so it seems highly degenerate? Or could we look at it as a limiting case of applying some deformation to the domain in case (1)? If the PDE with this domain is in fact valid, is there any issues I should be aware of due to the singular nature of the geometry?

Poisson/Laplace equation