no. 3-4
Ordinary Differential Equations
Non-homogeneous boundary conditions for a fourth-order diffusion equation
[Conditions aux limites non-homogènes pour une équation diffusive quantique stationnaire en une dimension]

Présenté par : Jean-Michel Bony

Amster, Pablo1 ; Jüngel, Ansgar2 ; Matthes, Daniel3
1 Departamento de Matemática, Cuidad Universitaria, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
2 Institut für Analysis und Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria
3 Departimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy
Comptes Rendus. Mathématique, Tome 346 (2008) no. 3-4, pp. 143-148.

L'existence des solutions classiques pour une équation élliptique non-linéaire d'ordre quatre en une dimension, qui apparaît dans la modélisation des semi-conducteurs quantiques, est démontrée pour une classe de conditions aux limites non-homogènes en utilisant la théorie du degré. En plus, des résultats de non-existence pour d'autres classes de conditions aux limites sont établis.

The existence of classical solutions to a one-dimensional non-linear fourth-order elliptic equation arising in quantum semiconductor modeling is proved for a class of non-homogeneous boundary conditions using degree theory. Furthermore, some non-existence results for other classes of boundary conditions are presented.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.12.001
Amster, Pablo 1 ; Jüngel, Ansgar 2 ; Matthes, Daniel 3

1 Departamento de Matemática, Cuidad Universitaria, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
2 Institut für Analysis und Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria
3 Departimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy 
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Amster, Pablo; Jüngel, Ansgar; Matthes, Daniel. Non-homogeneous boundary conditions for a fourth-order diffusion equation. Comptes Rendus. Mathématique, Tome 346 (2008) no. 3-4, pp. 143-148. doi : 10.1016/j.crma.2007.12.001. http://www.numdam.org/articles/10.1016/j.crma.2007.12.001/

[1] Ancona, M.; Iafrate, G. Quantum correction to the equation of state of an electron gas in a semiconductor, Phys. Rev. B, Volume 39 (1989), pp. 9536-9540

[2] Bleher, P.; Lebowitz, J.; Speer, E. Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Commun. Pure Appl. Math., Volume 47 (1994), pp. 923-942

[3] Caceres, M.; Carrillo, J.A.; Toscani, G. Long-time behavior for a nonlinear fourth order parabolic equation, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 1161-1175

[4] Derrida, B.; Lebowitz, J.; Speer, E.; Spohn, H. Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett., Volume 67 (1991), pp. 165-168

[5] Gualdani, M.P.; Jüngel, A.; Toscani, G. A nonlinear fourth-order parabolic equation with non-homogeneous boundary conditions, SIAM J. Math. Anal., Volume 37 (2006), pp. 1761-1779

[6] A. Jüngel, D. Matthes, The Derrida–Lebowitz–Speer–Spohn equation: existence, non-uniqueness, and decay rates of the solutions, SIAM J. Math. Anal., in press

[7] Jüngel, A.; Pinnau, R. Global non-negative solutions of a nonlinear fourth-oder parabolic equation for quantum systems, SIAM J. Math. Anal., Volume 32 (2000), pp. 760-777

[8] Lloyd, N. Degree Theory, Cambridge University Press, Cambridge, 1978

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