Communications in Mathematical Analysis

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

Irena Lasiecka and Daniel Toundykov

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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.

Article information

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

Dates
First available in Project Euclid: 7 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.cma/1270646497

Mathematical Reviews number (MathSciNet)
MR2551496

Zentralblatt MATH identifier
1189.35035

Subjects
Primary: 35B45: A priori estimates 35A05
Secondary: 74K20: Plates 93D15: Stabilization of systems by feedback 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE [See also 32C38, 58J15]

Keywords
plate non-dissipative non-monotone boundary condition shear feedback moment feedback semigroup hidden regularity microlocal a priori estimates

Citation

Lasiecka , Irena; Toundykov , Daniel. Semigroup Generation and ``hidden" Trace Regularity of a Dynamic Plate with Non-Monotone Boundary Feedbacks. Commun. Math. Anal. 8 (2010), no. 1, 109--144. https://projecteuclid.org/euclid.cma/1270646497


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