Transactions of the American Mathematical Society

Author(s): Avner Friedman; Juan J. L. Velázquez
Journal: Trans. Amer. Math. Soc. 349 (1997), 2537-2603.
MSC (1991): Primary 35J40
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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 $B_j$

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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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35J40

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

DOI: 10.1090/S0002-9947-97-01955-7
PII: S 0002-9947(97)01955-7
Keywords: Elliptic equations, boundary value problems, Liouville's theorem, Green's function
Received by editor(s): April 6, 1995
Copyright of article: Copyright 1997, American Mathematical Society

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