The Electropainting Problem with Overpotentials

Abstract

Existence, uniqueness and regularity are proved for the electropainting problem with over-potentials. The problem consists of finding a pair $\{ \varphi (x,t),h(x,t)\} $ such that $\varphi (x,t)$ is a time dependent family of harmonic functions, representing the electric potential in a domain, and $h(x,t)$ is related to the paint thickness on the part of the boundary being painted. The boundary condition on this part is $\varphi _n = G(\varphi ,h)$ where $\varphi _n $ is the inward normal derivative. h is determined from the history of the process. The assumption on the overpotential $\sigma (x)$ implies $h \geqq \sigma (x) > 0$ and thus the boundary condition is nondegenerate. We show that the process is monotone; there is no paint dissolution. Then we consider the explicit time discretization of the problem. Letting the time step shrink to zero leads to the above mentioned results. Then the $t \to \infty $ limit is considered, existence, uniqueness and regularity are proved. Moreover it is shown that th...