Sharp Observability Inequalities for the 1-D Plate Equation with a Potential |
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Authors: | Xiaoyu FU |
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Affiliation: | (1) College of Mathematics, Sichuan University, Chengdu, 610064, China;(2) Department of Mathematics, Indian Institute of Technology, Bombay, Mumbai, 400076, India |
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Abstract: | This paper deals with the problem of sharp observability inequality for the 1-D plate equation w tt + w xxxx + q(t, x)w = 0 with two types of boundary conditions w = w xx = 0 or w = w x = 0, and q(t, x) being a suitable potential. The author shows that the sharp observability constant is of order $exp left( {Cleft| q right|_infty ^{tfrac{2}
{7}} } right)$exp left( {Cleft| q right|_infty ^{tfrac{2}
{7}} } right) for ‖q‖∞ ≥ 1. The main tools to derive the desired observability inequalities are the global Carleman inequalities, based on a new point wise inequality for the fourth order plate operator. |
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Keywords: | Observability inequality Plate equation Point-wise estimate Carleman estimate |
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