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

DOI:
https://doi.org/10.1090/S0002-9947-97-01955-7

MathSciNet review:
1422604

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Abstract | References | Similar Articles | Additional Information

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.

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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

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