th order ordinary differential operators with general Wentzell boundary conditions

In some previous papers (see, e.g., [5], [6], [4]) we showed how to solve linear parabolic equations of the form Dtu = Au (A a second order elliptic operator) with boundary conditions of the form αAu+ β ∂u∂n + γu = 0 on ∂Ω, provided that β, γ ∈ C1(∂Ω), β > 0 on ∂Ω. Here we find the corresponding results for the fourth order operator A of the type Au := (au′′)′′, where we assume that

(A1) a ∈ C4[0, 1], a(x) > 0 in [0, 1], with general Wentzell boundary conditions of the type

(BC)j Au(j)+βj(au′′)′(j)+ γju(j) = 0 j = 0, 1,

A.