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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

$\left(1\right)\phantom{\rule{1.em}{0ex}}I\left(w\left(·\right)\right)={\int }_{0}^{\infty }f\left(w\left(s\right),\stackrel{˙}{w}\left(s\right),\phantom{\rule{1.em}{0ex}}\stackrel{¨}{w}\left(s\right)\right)ds,$
$\left(2\right)\phantom{\rule{1.em}{0ex}}w\in {A}_{x}=\left\{v\in {W}_{loc}^{2,1}\left(0,\infty \right):\phantom{\rule{1.em}{0ex}}\left(v\left(0\right),\stackrel{˙}{v}\left(0\right)\right)=\underline{x}\right\}·$

Here ${W}_{loc}^{2,1}\subset {C}^{1}$ denotes the Sobolev space of functions possessing a locally integrable second derivative, and $f=f\left(w,p,r\right)$ is a smooth function satisfying ${f}_{rr}\ge 0,$ $f\left(w,p,r\right)\ge {a|w|}^{\alpha }-{b|p|}^{\beta }+c{|r|}^{\gamma }-d;$ $a,b,c,d>0;$ $\alpha ,\gamma \in \left(1,\infty \right),$ $\beta \in \left[1,\infty \right),$ $\alpha >\beta ,$ $\gamma >\beta$ 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

##### MSC:
 74S30 Other numerical methods in solid mechanics 74A99 Generalities, axiomatics, foundations of continuum mechanics of solids 49J27 Optimal control problems in abstract spaces (existence) 49K27 Optimal control problems in abstract spaces (optimality conditions)