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

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 $Re\,\omega \geq0,\,\omega \neq0$ 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 $\varphi \equiv0$. 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) $Re\,\omega \geq0,\,\omega \neq0$ and (ii) $\omega =0$; both methods can be extended to other linear elliptic boundary value problems in a half plane.