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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Published in: | SIAM journal on mathematical analysis 1982-09, Vol.13 (5), p.746-757 |
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Main Author: | |
Format: | Article |
Language: | English |
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Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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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 $(x,y) \to (0,0)$. |
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ISSN: | 0036-1410 1095-7154 |
DOI: | 10.1137/0513051 |