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Proof of extensions of two conjectures on structural damping for elastic systems. (English) Zbl 0633.47025

Let A (the elastic operator) be a positive, self-adjoint operator with domain D(A) in the Hilbert space X, and let B (the dissipation operator) be another positive, selfadjoint operator satisfying \(\rho _ 1A^{\alpha}\leq B\leq \rho _ 2A^{\alpha}\) for some constants \(0<\rho _ 1<\rho _ 2<\infty\) and \(0\leq \alpha \leq 1\). Consider the operator \({\mathcal A}_ B=\left| \begin{matrix} 0\\ -A\end{matrix} \begin{matrix} I\\ - B\end{matrix}\right |\) (corresponding to the elastic model ẍ\(+B\dot x+Ax=0\) written as a first order system), which (once closed) is plainly the generator of a strongly continuous semigroup of contractions on the space \(E=D(A^{1/2})\times X\). We prove that if \(\leq \alpha \leq 1\), then such semigroup is also analytic (holomorphic) on a triangular sector of \({\mathbb{C}}\) containing the positive real axis. This established a fortiori two conjectures of Goong Chen and David L. Russell on structural damping for elastic systems, which referred to the case \(\alpha =\). Actually, in the special case \(\alpha =\) we prove a result stronger than the two conjectures, which yields analyticity of the semigroup over an explicitly identified range of spaces which includes E. This latter result has already proved in our previous effort on this problem. Here we provide a technically different and simplified proof of it. We also provide two conceptually and technically different proofs of our main result for \(\leq \alpha \leq 1\). Finally, we show that for \(0<\alpha <\), then the semigroup is not analytic.

MSC:

47D03 Groups and semigroups of linear operators
47E05 General theory of ordinary differential operators
74B05 Classical linear elasticity
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