International Conference
"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

Tkachev D.L.   Блохин А.М.  

The regularity of a solution and the well-posedness of an initial-boundary value problem for an elliptic system with a gradient quadratic nonlinearity

Reporter: Tkachev D.L.

In the last time, for the macroscopic description of the charge transport in semiconductors besides well-known drift-diffusion equations [1] and energy transport models[2] one becomes to use new hydrodynamical models [3]. These models are derived from the infinite system of moment equations by a suitable truncation procedure (the moment equations follow from the Boltzman transport equation). For justifying the stabilization method used for funding stationary solutions of the initial-boundary value problem (as a material basis we choose a planar silicon transistor MESFET (Metal Semiconductor Field Effect Transistor)) we have to prove that the obtained “limit” (in our case, elliptic) problem is well-posed. The essential feature of our problem is that the interior equations contain squares of gradients of the unknown functions. In the case when the right-hand side of the elliptic problem satisfies the condition that it grows as the “almost” uniform norm of the solution does (so-called “natural condition” [4]) we obtained the following two results: 1. The bounded solution of the problem has an additional smoothness and belongs to an intersection of Hölder and Sobolev spaces. 2. There exists a solution of the problem and it is unique under an additional assumption.   

This work was partially supported by RFBR (grant N 10-01-00320-a) and was done in the framework of the programs of the Russian Education Ministry “Russian scientific and educational personnel” 2009-2013 (grant N P1180) and “Development of the scientific potential of the High school” 2009-2011 (grant N 2.1.1/4591).

References
1. S. Selberherr. Analysis and Simulation of Semiconductor Devices, Wien, New York, Springer-Verlag, 1984.
2. D. Chen, E.C. Kan, U. Ravaioli, C-W. Shu, R. Dutton. An improved energy-transport model including nonparabolicity and non-maxwellian distribution effects, IEEE on Electron Device Letters, 13 (1992), 26-28.
3. A.M. Anile, V. Romano. Hydrodynamical modeling of charge carrier transport in semiconductors, MECCANICA, 35 (2000), 249-296.
4. S. Hilderbrandt, K.-O. Widman. Some regularity results for quasilinear elliptic systems of second order, Math. Z., 142 (1975), 67-86.