Fourth order ordinary differential operators with general Wentzell boundary conditions

We consider the fourth order ordinary differential operator Au:= (au″)″ with boundary conditions (eqution found) and one of uʹ(j); u″(j) vanishes for j = 0, 1: Here β0 \u3c 0 \u3c β1: Then A is essentially selfadjoint and bounded below on the Hilbert space H = L2(0, 1) ⊕ C2w, the completion of C[0, 1] under the inner product (eqution found) where wj:= (–1)j+1/Bj for j = 0, 1. Applications to partial differential equations are given