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The structure of extremals of a class of second order variational problems. (English) Zbl 0989.49003
Summary: We study the structure of extremals of a class of second-order variational problems without convexity, on intervals in \(R_+\). The problems are related to a model in thermodynamics introduced in [B. D. Coleman, M. Marcus and V. J. Mizel, Arch. Ration. Mech. Anal. 117, No. 4, 321-347 (1992; Zbl 0788.73015)]. We are interested in properties of the extremals which are independent of the length of the interval, for all sufficiently large intervals. As in [M. Marcus, Arch. Ration. Mech. Anal. 124, No. 1, 67-98 (1993; Zbl 0793.49019); Calc. Var. Partial Differ. Equ. 6, No. 2, 123-142 (1998; Zbl 0897.49010)] the study of these properties is based on the relation between the variational problem on bounded, large intervals and a limiting problem on \(R_+\). Our investigation employs techniques developed in [A. Leizarowitz and V. J. Mizel, Arch. Ration. Mech. Anal. 106, No. 2, 161-193 (1989; Zbl 0672.73010)]and [M. Marcus (loc. cit.)] along with turnpike techniques developed in [A. J. Zaslavski, J. Math. Anal. Appl. 194, No. 3, 660-696 (1995; Zbl 0860.49001); ibid. 198, No. 3, 893-921 (1996; Zbl 0881.49001)].

MSC:
49J10 Existence theories for free problems in two or more independent variables
49J45 Methods involving semicontinuity and convergence; relaxation
49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
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