Sharp observability inequalities for the 1-D plate equation with a potential

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 $\exp \left( {C\left\| q \right\|_\infty ^{\tfrac{2}{7}} } \right)$ 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.

Keywords

Observability inequality / Plate equation / Point-wise estimate / Carleman estimate

References

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