On the Structure of Solutions $\Delta ^2 u = \lambda u$ Which Satisfy the Clamped Plate Conditions on a Right Angle

Let u be a nontrivial solution of $\Delta ^2 u = \lambda u(\lambda > 0)$ on the quarter circle $\{ (x,y):0 < x,y,x^2 + y^2 < 1\} $ and suppose that \[ u(x,0) = u_y (x,0) = 0,\quad 0 < x < 1,\quad u(0,y) = u_x (0,y) = 0,\quad 0 < y < 1.\] We show then that on any ray through the origin $u(x,y)$ either vanishes identically or oscillates infinitely often as $(x,y) \to (0,0)$.