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Semigroup Generation and ``hidden" Trace Regularity of a Dynamic Plate with Non-Monotone Boundary Feedbacks

Commun. Math. Anal.
Volume 8, Number 1 (2010), 109 - 144

Semigroup Generation and ``hidden" Trace Regularity of a Dynamic Plate with Non-Monotone Boundary Feedbacks

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Abstract

The paper establishes well-posedness and semigroup generation for a linear dynamic plate equation with non-monotone boundary conditions. The lack of dissipation prevents applicability of the classical semigroup theory, approximation techniques, or energy methods. Investigation of such systems was originally motivated by applications, but due to the challenging nature of the problem had been essentially limited to 1-dimensional models. A more recent result [BeLa], though still dealing with a (1D) Euler-Bernoulli beam, showed how the wellposedness in absence of dissipativity can be approached using tools of microlocal analysis, potentially applicable in higher dimensions. This paper extends the later work to a two dimensional system. The main difficulties in the 2D setting arise from a substantially increased complexity of boundary operators, and the failure of the uniform Lopatinskii condition, which ultimately necessitates additional control on tangential components of the boundary traces. The latter issue is handled by introducing a suitably constructed boundary feedback which acts as the additional moment present on the boundary of the two-dimensional domain.