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Cooperation with: L. Recke (HumboldtUniversität zu Berlin)
Supported by: DFG: ``Zur Analysis von thermodynamischen Modellen des Stoff, Ladungs und Energietransports in heterogenen Halbleitern'' (Analysis of thermodynamic models for the transport of mass, charge and energy in heterogeneous semiconductors)
Description: We consider a stationary energy model for reactiondiffusion processes of electrically charged species with nonlocal electrostatic interaction. Such problems arise in electrochemistry as well as in semiconductor device and technology modeling (see, e.g., [2]). But, in contrast to [2] we now additionally take thermal effects into account. We consider a finite number of species X_{i}, (e.g., electrons, holes, dopants, interstitials, vacancies, dopantdefect pairs). Let and T be the electrostatic potential and the lattice temperature. We denote by the particle density of the ith species, its electrochemical potential and its charge number. The state equations are assumed to be given by the ansatz (see [1])
We consider a finite number of reversible reactions of the form The set of stoichiometric coefficients belonging to all reactions is denoted by .According to the massaction law the reaction rates are prescribed by Here u means the vector .For the particle flux densities j_{i} and the total energy flux density j_{e} we make the ansatz (see [1]) with conductivities , fulfilling where for all nondegenerated states u, T. For the fluxes and the generalized forces , the Onsager relations are fulfilled. The basic equations of the energy model contain n continuity equations for the considered species, the conservation law of the total energy and the Poisson equation,(1) 
(2) 
(3) 
If the boundary values , ,are constants, and , , correspond to a simultaneous equilibrium of all reactions
and if , are arbitrarily given, then there exists a unique solution of . is a thermodynamic equilibrium of (2), (3). The operator F is continuously differentiable and the linearization turns out to be an injective Fredholm operator of index zero. This follows from results in [5] and from a regularity result of Gröger in [4] for systems of elliptic equations with mixed boundary conditions. Therefore we can apply the Implicit Function Theorem and obtain that for near ,f near f^{*} and g near g^{*} the equation has a unique solution near . Thus, near there is a locally unique Hölder continuous solution of (2), (3). For details and the precise assumptions of our investigations see [3].Our local existence and uniqueness result for the stationary energy model (2), (3) works in two space dimensions. But let us note that in our model equations crossterms with respect to all species and temperature are involved.
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