Abstract
In this paper we consider the problem of minimizing
in the set \(ar A\) of absolutely continuous functions y(·): [a, b] → ℝ satisfying the end conditions
where α is a prescribed constant. In (1), [a, b] is a bounded interval, “′ ” denotes differentiation with respect to x, and the integrand \(ar f = ar f(x,y,p)\) is assumed to be smooth (C 3) and nonnegative. In addition, \(ar f\) is assumed to satisfy:
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
T. S. Angell, “A note on approximation of optimal solutions of free problems of the calculus of variations”, Rend. Circ. Mat. Palermo 2 (1979), 258–272.
J. M. Ball, J. C. Currie, & P. J. Olver, “Null Lagrangians, weak continuity, and variational problems of arbitrary order”, J. Funct. Anal. 41 (1981), 135–174.
J. M. Ball & G. Knowles, “A numerical method for detecting singular minimizers”, Numer. Math. 51 (1987), 181–197.
J. Ball & V. J. Mizel, “Singular minimizers for regular one-dimensional problems in the calculus of variations”, Bull. Am. Math. Soc. 11 (1984), 143–146.
J. Ball & V. J. Mizel, “One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation”, Arch. Rational Mech. Anal. 90 (1985), 325–388.
L. Cesari, Optimization-Theory and Applications, Berlin Heidelberg New York Tokyo: Springer-Verlag, 1983.
F. H. Clarke & R. B. Vinter, “Regularity properties of solutions to the basic problem in the calculus of variations”, Trans. Am. Math. Soc. 291 (1985), 73–98.
A. M. Davie, “Singular minimizers in the calculus of variations in one dimension”, Arch. Rational Mech. Anal. (1987) (to appear).
A. C. Heinricher, “A singular stochastic control problem arising from a deterministic problem with non-Lipschitzian minimizers”, Ph. D. Dissertation, Department of Mathematics, Carnegie-Mellon University (1986).
A. C. Heinricher & V. J. Mizel, “A stochastic control problem with different value functions for singular and absolutely continuous control”, Proc. 1986 IEEE Conf. Decision and Control, Athens, Greece.
A. C. Heinricher & V. J. Mizel, “A new example of the Lavrentiev phenomenon”, SIAM Journal for Control and Optimization, to appear.
M. R. Hestenes, Calculus of Variations and Optimal Control Theory, Krieger Publishing, 1980.
M. Lavrentiev, “Sur quelques problèmes du calcul des variations”, Ann. Mat. Pura Appl. 4 (1926), 107–124.
P. D. Loewen, “On the Lavrentiev Phenomenon”, Can. Math. Bull. 30 (1987), 102108.
J. D. Logan, Invariant Variational Principles, New York London: Academic Press, 1977.
E. J. Mcshane, “Existence theorems for problems in the calculus of variations”, Duke Math. J. 4 (1938), 132–156.
B. Manià, Sorpra un esempio di Lavrentieff, Bull. Un. Mat. Ital. 13 (1934), 146–153.
M. Marcus & V. Mizel, “Absolute continuity on tracks and mappings of Sobolev spaces”, Arch. Rational Mech. Anal. 45 (1972), 294–320.
E. Noether, “Invariante Variationsprobleme”, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 11 (1918), 235–257.
E. Noether, “Invariant Variational Problems”, Transport Theory of Statis. Physics 1, 186–207 (1971) ( Translation by M. A. Tavel of the original article).
P. J. Olver, Applications of Lie Groups to Differential Equations, Berlin Heidelberg New York Tokyo: Springer-Verlag, 1986.
L. Tonelli, “Sugli integrali del calcolo delle variazioni in forma ordinaria”, Ann. R. Scuola Norm. Sup. Pisa 3 (1934), 401–450 (in L. Tonelli Opere Scelte vol. III # 105, Edizioni Cremonese, Roma, 1961 ).
CH. de la Vallée Poussin, “Sur l’intégrale de Lebesgue”, Trans. Am. Math. Soc.16 (1915), 435–501.
Author information
Authors and Affiliations
Additional information
Dedicated toJames B. Serrin
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Heinricher, A.C., Mizel, V.J. (1989). The Lavrentiev Phenomenon for Invariant Variational Problems. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-83743-2_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50917-2
Online ISBN: 978-3-642-83743-2
eBook Packages: Springer Book Archive