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

Abstract

Consider an elliptic equation ω Δ φ − Δ 2 φ = 0 \omega \Delta \varphi -\Delta ^2\varphi =0 in the half plane { ( x , y ) , − ∞ > x > ∞ , y > 0 } \{(x,\,y),\,-\infty >x>\infty ,\,y>0\} with boundary conditions φ = φ y = 0 \varphi =\varphi _y=0 if y = 0 , x > 0 y=0,\,x>0 and B j φ = 0 B_j\varphi =0 if y = 0 , x > 0 y=0,\,x>0 where B j B_j ( j = 2 , 3 ) (j=2,3) are second and third order differential operators. It is proved that if R e ω ≥ 0 , ω ≠ 0 Re\,\omega \geq 0,\,\omega \neq 0 and, for some ε > 0 \varepsilon >0 , | φ | ≤ C r α |\varphi |\leq Cr^\alpha if r = x 2 + y 2 → ∞ , | φ | ≤ C r β r=\sqrt {x^2+y^2}\to \infty ,\quad |\varphi |\leq Cr^\beta if r → 0 r\to 0 where α = n + 1 2 − ε , β = n + 1 2 + ε \alpha =n+\frac {1}{2}-\varepsilon \,,\quad \beta =n+\frac {1}{2}+\varepsilon for some nonnegative integer n n , then φ ≡ 0 \varphi \equiv 0 . Results of this type are also established in case ω = 0 \omega =0 under different conditions on α \alpha and β \beta ; furthermore, in one case B 3 φ B_3\varphi has a lower order term which depends nonlocally on φ \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) R e ω ≥ 0 , ω ≠ 0 Re\,\omega \geq 0,\,\omega \neq 0 and (ii) ω = 0 \omega =0 ; both methods can be extended to other linear elliptic boundary value problems in a half plane.