A mixed problem for electrostatic potential in semiconductors

In this article we deal with the solution in Ω ⊂ R 2 of the quasi linear equation −Δ u = f ( x, y, u ( x, y )) subject to mixed boundary data and representing Gauss' law in a semiconductor device, where u and f are, respectively, the electrostatic potential and the space charge density after a suitable scaling. In the following we consider the associated variational problem of finding in a suitable subspace of H 1 (Ω) the minimum of the functional \documentclass{article}\pagestyle{empty}\begin{document}$ J(u)\, = \,\int {_\Omega } (\frac{1}{2}\left| {\nabla u\left| {^2 \, - \,{\cal F}(x,y,u)\,d\Omega,} \right.} \right. $\end{document} , where \documentclass{article}\pagestyle{empty}\begin{document}$ {\cal F}(x,y,u)\, = \,\int f (x,y,\xi)\,d\xi, $\end{document} and we prove existence and uniqueness of a weak solution according to the technique of Convex Analysis. The numerical study is then carried on by a piecewise linear finite element approximation, which is proved to converge in the H 1 ‐norm to the exact solution of the variational problem; some numerical examples are also included. © 1994 John Wiley & Sons, Inc.