A Free Boundary Problem Arising in Electrophotography: Solutions with Connected Toner Region

A free boundary problem that arises in the development of a photocopy is studied. The electric potential $ - u$ satisfies the equation $\Delta u = 1$ in the toner region and $\Delta u = 0$ elsewhere. It is shown that $C^{1 + \alpha } $ smoothness of the free boundary would imply the $C^{2 + \alpha } $ smoothness of the solution up to both sides of the free boundary. Using this fact, the existence of a solution with connected it is proven that toner region with $\frac{{\partial u}}{{\partial n}} = 0$ on the free boundary when the electrical charge length is “small.”