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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 $(1)\quad I(w(\cdot))=\int^{\infty}_{0}f(w(s),\dot w(s),\quad \ddot w(s))ds,$ $(2)\quad w\in A_ x=\{v\in W^{2,1}_{loc}(0,\infty):\quad (v(0),\dot v(0))=\underline x\}.$ Here $$W^{2,1}_{loc}\subset C^ 1$$ denotes the Sobolev space of functions possessing a locally integrable second derivative, and $$f=f(w,p,r)$$ is a smooth function satisfying $$f_{rr}\geq 0,$$ $$f(w,p,r)\geq a| w|^{\alpha}-b| p|^{\beta}+c| r|^{\gamma}-d;$$ $$a,b,c,d>0;$$ $$\alpha,\gamma \in (1,\infty),$$ $$\beta \in [1,\infty),$$ $$\alpha >\beta,$$ $$\gamma >\beta$$ 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

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74A99 Generalities, axiomatics, foundations of continuum mechanics of solids 49J27 Existence theories for problems in abstract spaces 49K27 Optimality conditions for problems in abstract spaces
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##### References:
  Aubry, S., & P. Y. Le Daeron, The discrete Frenkel-Kontorova model and · Zbl 1237.37059  Artstein, Z., & A. Leizarowitz, Tracking periodic signals with overtaking criterion, IEEE Trans. on Autom. Control AC 30 (1985), 1122–1126. · Zbl 0576.93035 · doi:10.1109/TAC.1985.1103851  Brock, W. A., A. Haurie, On existence of overtaking optimal trajectories over an infinite time · Zbl 0367.49003 · doi:10.1287/moor.1.4.337  Carlson, D., On the existence of catching-up optimal solutions for Lagrange problems defined on unbounded interva · Zbl 0573.49003 · doi:10.1007/BF00940757  Coleman, B. D., Necking and drawing in polymeric fibers under tension, Ar · Zbl 0535.73016 · doi:10.1007/BF00282158  Coleman, B. D., On the cold drawing of polymers, Comp. · Zbl 0634.73030 · doi:10.1016/0898-1221(85)90137-3  Chou, W., & R. J. Duffin, An additive eigenvalue problem of physics related to linear programm · Zbl 0639.65033 · doi:10.1016/0196-8858(87)90022-4  Cahn, J. W., & J. E. Milliard, Free energy of a nonuniform system. I, Inter-facial free · doi:10.1063/1.1744102  Carr, J., M. E., Gurtin & M. Slemrod, Structural phase transitions on a finite interval, Ar · Zbl 0564.76075 · doi:10.1007/BF00280031  Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies # 105, Princeton U. Press, Princeton 1983. · Zbl 0516.49003  Griffiths, R. B., & W. Chou, Effective potentials: a new approach and new results for one-dimensional systems with competing length sc · doi:10.1103/PhysRevLett.56.1929  Leizarowitz, A., Infinite horizon autonomous systems with unbounded co · Zbl 0591.93039 · doi:10.1007/BF01442197  Lions, J. L., & E. Magenes, Non-homogeneous boundary value problems and Applications I, Grundlehren # 181, Springer, Berlin, 1972. · Zbl 0223.35039  Mather, J. N., More Denjoy minimal sets for area preserving diffeomorphisms, · Zbl 0597.58015 · doi:10.1007/BF02567431  Morrey, C. B., Jr., Multiple integrals in the calculus of variations, Grundlehren # 130, Springer, New York, 1966.  Nabarro, F. R. N., Theory of crystal dislocations, Clarendon Press, Oxford, 1967.  Tonelli, L., Sugli integrali del calcolo delle variazioni in forma ordinaria · Zbl 0010.11903  van der Waals, J. D., The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch), verhandel. Konink. Akad. Weten. Amsterdam (Sec. 1) 1 (1893).  von Weizsacker, C. C., Existence of optimal programs of accumulation for an infinite hor · doi:10.2307/2296054
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