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 th...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 1982-09, Vol.13 (5), p.746-757
Main Author: Coffman, Charles V.
Format: Article
Language:English
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Summary: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 $(x,y) \to (0,0)$.
ISSN:0036-1410
1095-7154
DOI:10.1137/0513051