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On the holomorphic property of the semigroup associated with linear elastic systems with structural damping. (English) Zbl 0636.73047

G. Chen and D. L. Russell [(*) A mathematical model for linear elastic systems with structural damping, MRC Technical summary Report 2089, Math. Res. Center. Univ. Wisconsin-Madison (1980)] studied the following linear elastic systems with structural damping

y ¨+By ˙+Ay=0,y(0)=y 0 ,y ˙(0)=y 1 ,

where · means d dt, y,y 0 ,y 1 H, a Hilbert space with inner product (·,·) and associated with ·, A and B are positive definite self-adjoint unbounded linear operators on H and B is A 1/2 -bounded. Letting x 1 =A 1/2 y, x 2 =y ˙, we get the equivalent first order linear systems

d dtx 1 x 2 =0A 1/2 -A 1/2 -Bx 1 x 2 =𝒜x 1 x 2 x 1 (0)=A 1/2 y 0 ,x 2 (0)=y 1 ·

G. Chen and D. L. Russel (*) have proved that 𝒜=0A 1/2 -A 1/2 -B generates a holomorphic semigroup, if some addition conditions are satisfied. Moreover, they still conjectured that if A and B are positive definite self-adjoint unbounded linear operators with the domain 𝒟(B)𝒟(A 1/2 ), such that

(I)β 1 (A 1/2 x,x)(Bx,x)β 2 (A 1/2 x,x),x𝒟(A 1/2 ),


(II)β 1 (Ax,x)(B2x,x)β 2 (Ax,x),x𝒟(A),

then 𝒜=0A 1/2 -A 1/2 -B should generate a holomorphic semi-group, where β 1 and β 2 are positive constant numbers with β 1 β 2 . Also some partial results for conjecture (I) and (II) are shown in (*). Recently, the author gave an answer to the conjecture (I) affirmatively [A problem for linear elastic systems with structural damping (to appear)]. In this paper we will show that conjecture (II) is also true.

74H45Vibrations (dynamical problems in solid mechanics)
47D03(Semi)groups of linear operators
47B25Symmetric and selfadjoint operators (unbounded)
20M05Free semigroups, generators and relations, word problems