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A steady state potential flow model for semiconductors. (English) Zbl 0808.35150
The paper concerns the hydrodynamic equations modelling a unipolar semiconductor. Here the unknown functions are \(\rho\) = electron density, \(u=\{ u_ 1, u_ 2, u_ 3\}\) = electron velocity and \(\varphi\) = electrostatic potential. Under the assumption of potential flow \(u= - \nabla\psi\) (\(\psi\) = velocity potential) the equations take the form \[ {\textstyle {1\over 2}}| \nabla\psi |^ 2+ h(\rho)= \varphi+ {\textstyle {\psi\over\tau}}, \qquad \text{div}(\rho \nabla\psi)=0, \quad \Delta\varphi= \rho-C \] which have to be considered in a bounded domain \(\Omega\subset \mathbb{R}^ 3\), where \(h\) = given function, \(\tau= \text{const}>0\) relaxation time and \(C= C(x)\) \((x\in\Omega)\) given doping profile.
By the aid of Schauder’s fixed point theorem the author prove the existence of a solution \((\psi,\rho, \varphi)\in C^{2,\delta} (\overline{\Omega})\times W^{2,q}(\Omega)\times W^{2,q} (\Omega)\) \((1\leq q<\infty)\) provided \(\Omega\) is a bounded convex \(C^{2,\delta}\)-domain in \(\mathbb{R}^ 3\), \(\psi\), \(\rho\), \(\varphi\) are subject to Dirichlet conditions along the whole boundary \(\partial\Omega\) and an appropriate smallness condition on the Dirichlet datum of \(\psi\) is satisfied. This smallness condition implies ellipticity of the problem under consideration which is equivalent to the case of subsonic flow of electrons in the semiconductor device.
Reviewer: J.Naumann (Berlin)

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
78A35 Motion of charged particles
76G25 General aerodynamics and subsonic flows
35Q35 PDEs in connection with fluid mechanics
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