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

DOI:
https://doi.org/10.1090/S0002-9947-1990-0970266-6

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 , . 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 , 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 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 exhibit pathologies not present in the case.

**[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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DOI:
https://doi.org/10.1090/S0002-9947-1990-0970266-6

Article copyright:
© Copyright 1990
American Mathematical Society