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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

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
Full Text: DOI
[1] Aubry, S., & P. Y. Le Daeron, The discrete Frenkel-Kontorova model and · Zbl 1237.37059
[2] 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
[3] 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
[4] Carlson, D., On the existence of catching-up optimal solutions for Lagrange problems defined on unbounded interva · Zbl 0573.49003 · doi:10.1007/BF00940757
[5] Coleman, B. D., Necking and drawing in polymeric fibers under tension, Ar · Zbl 0535.73016 · doi:10.1007/BF00282158
[6] Coleman, B. D., On the cold drawing of polymers, Comp. · Zbl 0634.73030 · doi:10.1016/0898-1221(85)90137-3
[7] 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
[8] Cahn, J. W., & J. E. Milliard, Free energy of a nonuniform system. I, Inter-facial free · doi:10.1063/1.1744102
[9] Carr, J., M. E., Gurtin & M. Slemrod, Structural phase transitions on a finite interval, Ar · Zbl 0564.76075 · doi:10.1007/BF00280031
[10] 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
[11] 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
[12] Leizarowitz, A., Infinite horizon autonomous systems with unbounded co · Zbl 0591.93039 · doi:10.1007/BF01442197
[13] Lions, J. L., & E. Magenes, Non-homogeneous boundary value problems and Applications I, Grundlehren # 181, Springer, Berlin, 1972. · Zbl 0223.35039
[14] Mather, J. N., More Denjoy minimal sets for area preserving diffeomorphisms, · Zbl 0597.58015 · doi:10.1007/BF02567431
[15] Morrey, C. B., Jr., Multiple integrals in the calculus of variations, Grundlehren # 130, Springer, New York, 1966.
[16] Nabarro, F. R. N., Theory of crystal dislocations, Clarendon Press, Oxford, 1967.
[17] Tonelli, L., Sugli integrali del calcolo delle variazioni in forma ordinaria · Zbl 0010.11903
[18] 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).
[19] von Weizsacker, C. C., Existence of optimal programs of accumulation for an infinite hor · doi:10.2307/2296054
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