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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Liouville type theorems for fourth order elliptic equations in a half plane

Authors: Avner Friedman and Juan J. L. Velázquez
Journal: Trans. Amer. Math. Soc. 349 (1997), 2537-2603
MSC (1991): Primary 35J40
MathSciNet review: 1422604
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Abstract: Consider an elliptic equation $\omega \Delta\varphi -\Delta ^2\varphi =0$ in the half plane $\{(x,\,y),\,-\infty <x<\infty ,\,y>0\}$ with boundary conditions $\varphi =\varphi _y=0$ if $y=0,\,x>0$ and $B_j\varphi =0$ if $y=0,\,x<0$ where $B_j$ $(j=2,3)$ are second and third order differential operators. It is proved that if $Re\,\omega \geq0,\,\omega \neq0$ and, for some $\varepsilon >0$, $|\varphi |\leq Cr^\alpha $ if $r=\sqrt {x^2+y^2}\to \infty ,\quad |\varphi |\leq Cr^\beta $ if $r\to 0$ where $\alpha =n+\frac {1}{2}-\varepsilon \,,\quad \beta=n+\frac {1}{2}+\varepsilon $ for some nonnegative integer $n$, then $\varphi \equiv0$. Results of this type are also established in case $\omega =0$ under different conditions on $\alpha $ and $\beta $; furthermore, in one case $B_3\varphi $ has a lower order term which depends nonlocally on $\varphi $. Such Liouville type theorems arise in the study of coating flow; in fact, they play a crucial role in the analysis of the linearized version of this problem. The methods developed in this paper are entirely different for the two cases (i) $Re\,\omega \geq0,\,\omega \neq0$ and (ii) $\omega =0$; both methods can be extended to other linear elliptic boundary value problems in a half plane.

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Additional Information

Avner Friedman
Affiliation: University of Minnesota, Institute for Mathematics and its Applications, Minneapolis, Minnesota 55455

Juan J. L. Velázquez
Affiliation: Departamento de Matematica Aplicada, Universidad Complutense, Facultad de Matematicas 28040, Madrid, Spain

Keywords: Elliptic equations, boundary value problems, Liouville's theorem, Green's function
Received by editor(s): April 6, 1995
Article copyright: © Copyright 1997 American Mathematical Society