zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 ),

or

(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.

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