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

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- by Avner Friedman and Juan J. L. Velázquez PDF
- Trans. Amer. Math. Soc.
**349**(1997), 2537-2603 Request permission

## 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 $Re \omega \geq 0, \omega \neq 0$ 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 $\varphi \equiv 0$. 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) $Re \omega \geq 0, \omega \neq 0$ and (ii) $\omega =0$; both methods can be extended to other linear elliptic boundary value problems in a half plane.## References

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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
- MR Author ID: 289301
- Received by editor(s): April 6, 1995
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**349**(1997), 2537-2603 - MSC (1991): Primary 35J40
- DOI: https://doi.org/10.1090/S0002-9947-97-01955-7
- MathSciNet review: 1422604