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, self-adjoint operator satisfying:
ρ1Aα≤ B ≤ ρ2Aα for some constants 0 < ρ1< ρ2< ∞ and 0 ≤ α ≤ 1. Consider the
operator
(corresponding to the elastic model ẍ+ Bẋ+ 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(A1∕2) × X. We prove that if
1∕2 ≤ α ≤ 1, then such semigroup is also analytic (holomorphic) on a triangular
sector of 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 α = 1∕2. Actually, in the special case α = 1∕2 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 was 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
1∕2 ≤ α ≤ 1. Finally, we show that for 0 < α < 1∕2 the semigroup is not
analytic.
