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Transactions of the American Mathematical Society

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A regularity theory for variational problems with higher order derivatives

Authors: F. H. Clarke and R. B. Vinter
Journal: Trans. Amer. Math. Soc. 320 (1990), 227-251
MSC: Primary 49A21; Secondary 49B21
MathSciNet review: 970266
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Abstract: We consider problems in the calculus of variations in one independent variable and where the Lagrangian involves derivatives up to order $ N$, $ N \ge 1$. Existence theory supplies mild hypotheses under which there are minimizers for such problems, but they need to be strengthened for standard necessary conditions to apply.

For problems with $ N > 1$, this paper initiates investigation of regularity properties, and associated necessary conditions, which obtain strictly under the hypotheses of existence theory. It is shown that the $ N$th derivative of a minimizer is locally essentially bounded off a closed set of zero measure, the set of "points of bad behaviour". Additional hypotheses are shown to exclude occurrence of points of bad behaviour. Finally a counter example suggests respects in which problems with $ N > 1$ exhibit pathologies not present in the $ N = 1$ case.

References [Enhancements On Off] (What's this?)

  • [1] L. D. Berkovitz, Lower semicontinuity of integral functionals, Trans. Amer. Math. Soc. 192 (1974), 51-57. MR 0348582 (50:1079)
  • [2] L. Cesari, Optimization--Theory and applications, Springer-Verlag, New York, 1983. MR 688142 (85c:49001)
  • [3] F. H. Clarke, Optimization and nonsmooth analysis, Wiley Interscience, New York, 1983. (Reprinted 1989 by CRM, University of Montreal, POB 6128-A, Montreal (Qc) Canada H3C 3J7.) MR 709590 (85m:49002)
  • [4] F. H. Clarke and P. D. Loewen, An intermediate existence theory in the calculus of variations, Centre de Recherches Mathématiques, Report CRM-1418, April 1987, Ann. Scuola Norm. Sup. (to appear). MR 1052732 (91c:49061)
  • [5] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. 289 (1985), 73-98. MR 779053 (86h:49020)
  • [6] -, Existence and regularity in the small in the calculus of variations, J. Differential Equations 59 (1985), 336-354. MR 807852 (87a:49014)
  • [7] -, On conditions under which the Euler equation or the maximum principle hold, Appl. Math. Optim. 12 (1984), 73-79. MR 756513 (85m:49051)
  • [8] -, Regularity of solutions to variational problems with polynomial Lagrangians, Bull. Polish Acad. Sci. 34 (1986), 73-81. MR 850317 (87j:49042)
  • [9] R. T. Rockafellar, Existence theorems for general control problems of Bolza and Lagrange, Adv. in Math. 15 (1975), 312-333. MR 0365273 (51:1526)
  • [10] D. R. Smith, Variational methods in optimization, Prentice-Hall, Englewood Cliffs, N. J., 1974. MR 0346616 (49:11341)
  • [11] L. Tonelli, Sur une méthode directe du calcul des variations, Rend. Circ. Math. Palermo 39 (1915), 233-264. Also appears in Opere scelte, vol. 2, Cremonese, Rome, 1961, pp. 289-333.
  • [12] -, Fondamenti di calcolo delle variazioni, vols. 1, 2, Zanichelli, Bologna, 1921, 1923.

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