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: Consider an elliptic equation in the half plane with boundary conditions if and if where are second and third order differential operators. It is proved that if and, for some , if if where for some nonnegative integer , then . Results of this type are also established in case under different conditions on and ; furthermore, in one case has a lower order term which depends nonlocally on . 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) and (ii) ; 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

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

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