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 ?Δu = 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 H¹(Ω) the minimum of the functional $ J(u)\, = \,\int {_\Omega } (\frac{1}{2}\left| {\nabla u\left| {^2 \, - \,{\cal F}(x,y,u)\,d\Omega,} \right.} \right. $, where $ {\cal F}(x,y,u)\, = \,\int f (x,y,\xi)\,d\xi, $ 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 H¹-norm to the exact solution of the variational problem; some numerical examples are also included. © 1994 John Wiley & Sons, Inc.