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J. 29, 560-607 (1950). 22] Vasileva, A. , Butuzow, V. : Singularly Perturbed Equations in the Critical Case. MRC-TSR2039, Math. Res. , 1980. 23] Vasileva, A. , Stelmakh, V. : Singularly Disturbed Systems of the Theory of Semiconductor Devices. USSR Comput. Math. Phys. 17,48-58 (1977). 24] Zimam, J. : Electrons and Phonons. London: Clarendon Press 1963. 1 Preliminaries In this chapter we present existence, regularity, uniqueness and continuousdependence-on-data results for the basic semiconductor device equations in the stationary case.

We now prove the existence and uniqueness theorem for small applied voltages. 2) hold. Then, if lUI < a holds for some sufficiently small a, the problem (SD) has a locally unique solution (Iff*( U), u*( U), v*( U)) E (H2(Q))3, which satisfies (1ff*(0), u*(O), v*(O)) = (Iff e' I, 1) and which depends continuously differentiably on U when regarded as map from {U E IW I lUI < a} into (H2(Q))3. 8) as operator equation G«({J, y, z, U) = O. 9) The domain of G is the Cartesian product of an open subset A of (H'§(Q))3, where H'§(Q):= {({JEH2(Q)lo({J/ovIDQN = ({JlaQD = O}, and a sphere SuJO) s; IW.

Grad v. 21) The parameters A and J are determined by the doping, length, material and temperature of the device under consideration. Both parameters are small (compared tQ 1) in practical situations, but their impact on the solutions is entirely different. A2 multiplies the highest order derivatives of the potential and therefore we expect it to directly influence the variation of the solutions. This conjecture will be totally confirmed later on. Here we only remark that a problem, in which a small parameter multiplies a derivative of highest order, is - in the mathematical literature - called a singular perturbation problem.

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