Linear Reduction and Homotopy Control for Steady Drift-Diffusion Systems in Narrow Convex Domains
Joseph W. Jerome
TL;DR
The paper addresses the existence of steady drift-diffusion solutions in narrow convex domains without assuming Boltzmann statistics or Einstein relations. It develops a continuation framework based on a differentiable mapping $F:\mathcal{H}\times[0,1]\to\mathcal{G}$ with Lipschitz derivatives $F_w$ and $F_\mu$ and a bounded approximate inverse, yielding a solution curve $h_\lambda$ that reaches the nonlinear target from a linear baseline. Applied to a Poisson–Nernst–Planck–style drift-diffusion model with potential-driven transport, the work defines $u$, $n$, $p$ and proves Fréchet differentiability, Lipschitz bounds, and the existence of a bounded inverse for $F_h$ on a small-width domain, enabling a robust predictor–corrector Newton framework. Nonnegativity of ion densities is established under mild boundary conditions, and the results provide a rigorous analytic foundation for numerical continuation and Newton-type solvers in narrow domains.
Abstract
This article develops and applies results, originally introduced in earlier work, for the existence of homotopy curves, terminating at a desired solution. We describe the principal hypotheses and results in section two; right inverse approximation is at the core of the theory. We apply this theory in section three to the basic drift-diffusion equations. The carrier densities are not assumed to satisfy Boltzmann statistics and the Einstein relations are not assumed. By proving the existence of the homotopy curve, we validate the underlying computational framework of a predictor/corrector scheme, where the corrector utilizes an approximate Newton method. The analysis depends on the assumption of domains of narrow width. However, no assumption is made regarding the domain diameter.