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 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.

**1.**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I*, Comm. Pure Appl. Math.**12**(1959), 623–727. MR**0125307****2.**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II*, Comm. Pure Appl. Math.**17**(1964), 35–92. MR**0162050****3.**Avner Friedman and Juan J. L. Velázquez,*The analysis of coating flows near the contact line*, J. Differential Equations**119**(1995), no. 1, 137–208. MR**1334490**, 10.1006/jdeq.1995.1086**4.**Avner Friedman and Juan J. L. Velázquez,*The analysis of coating flows in a strip*, J. Differential Equations**121**(1995), no. 1, 134–182. MR**1348539**, 10.1006/jdeq.1995.1125**5.**A. Friedman and J.J.L. Velázquez,*Time-dependent coating flows in a strip,*Trans. Amer. Math. Soc., to appear.**6.**V. A. Kondrat′ev,*Boundary value problems for elliptic equations in domains with conical or angular points*, Trudy Moskov. Mat. Obšč.**16**(1967), 209–292 (Russian). MR**0226187****7.**J.-L. Lions and E. Magenes,*Problèmes aux limites non homogènes et applications. Vol. 1*, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR**0247243****8.**V. G. Maz′ja and B. A. Plamenevskiĭ,*Estimates in 𝐿_{𝑝} and in Hölder classes, and the Miranda-Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary*, Math. Nachr.**81**(1978), 25–82 (Russian). MR**0492821****9.**V. G. Maz′ja and B. A. Plamenevskiĭ,*The coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points*, Math. Nachr.**76**(1977), 29–60 (Russian). MR**0601608**

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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:
http://dx.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