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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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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