zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
One-dimensional infinite-horizon variational problems arising in continuum mechanics. (English) Zbl 0672.73010
The authors study a variational problem for real valued functions defined on an infinite semiaxis of the line. They seek a “minimal solution” to the following problem: minimize the functional $$ (1)\quad I(w(\cdot))=\int\sp{\infty}\sb{0}f(w(s),\dot w(s),\quad \ddot w(s))ds, $$ $$ (2)\quad w\in A\sb x=\{v\in W\sp{2,1}\sb{loc}(0,\infty):\quad (v(0),\dot v(0))=\underline x\}. $$ Here $W\sp{2,1}\sb{loc}\subset C\sp 1$ denotes the Sobolev space of functions possessing a locally integrable second derivative, and $f=f(w,p,r)$ is a smooth function satisfying $f\sb{rr}\ge 0,$ $f(w,p,r)\ge a\vert w\vert\sp{\alpha}-b\vert p\vert\sp{\beta}+c\vert r\vert\sp{\gamma}-d;$ $a,b,c,d>0;$ $\alpha,\gamma \in (1,\infty),$ $\beta \in [1,\infty),$ $\alpha >\beta,$ $\gamma >\beta$ and $\underline x$ is a given value. The organization of the paper is as follows: In Section 1 the connection of the variational problem (1),(2) with the equilibrium problem of a long slender bar of polymetric material under tension is discussed. In Section 2 a fixed endpoint variational problem is analyzed. In Section 3 a criterion is given for the solution of (1),(2) to be minimal. Section 4 gives the proof the existence of a minimal energy solution. The Section 5 is containing the main result of the paper: there always exists a periodic minimal solution of (1),(2). In Section 7 an interesting analytic result is established.
Reviewer: I.Ecsedi

74S30Other numerical methods in solid mechanics
74A99Generalities, axiomatics, foundations of continuum mechanics of solids
49J27Optimal control problems in abstract spaces (existence)
49K27Optimal control problems in abstract spaces (optimality conditions)
Full Text: DOI