Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Existence theorems for parametric problems in the calculus of variations and approximation
HTML articles powered by AMS MathViewer

by Robert M. Goor
Trans. Amer. Math. Soc. 223 (1976), 347-365
DOI: https://doi.org/10.1090/S0002-9947-1976-0425716-3

Abstract:

In this paper, we investigate the parametric growth condition which arises in connection with existence theorems for parametric problems of the calculus of variations. In particular, we study conditions under which the length of a curve is dominated in a suitable sense by its “cost". We show that we may restrict our attention to local growth conditions on a particular set. Then we link the growth conditions to a certain approximation problem on this set. Finally, we prove that under suitable topological restrictions related to dimension theory, the local and global problems can be solved.
References
    L. Cesari, Existence theorems for weak and usual solutions in Lagrange problems with unilateral constraints. I, II, Trans. Amer. Math. Soc. 124 (1966), 369-412, 413-430. MR 34 # 3392; # 3393.
  • Lamberto Cesari, Closure theorems for orientor fields and weak convergence, Arch. Rational Mech. Anal. 55 (1974), 332–356. MR 350589, DOI 10.1007/BF00250438
  • Lamberto Cesari, Rectifiable curves and the Weierstrass integral, Amer. Math. Monthly 65 (1958), 485–500. MR 131504, DOI 10.2307/2308574
  • Lamberto Cesari, Seminormality and upper semicontinuity in optimal control, J. Optim. Theory Appl. 6 (1970), 114–137. MR 270248, DOI 10.1007/BF00927046
  • —, Surface area, Ann. of Math. Studies, no. 35, Princeton Univ. Press, Princeton, N.J., 1956. MR 17, 596.
  • George M. Ewing, Calculus of variations with applications, W. W. Norton & Co. Inc., New York, 1969. MR 0242032
  • Robert M. Goor, Gradient approximation of vector fields, J. Approximation Theory 12 (1974), 385–395. MR 355440, DOI 10.1016/0021-9045(74)90082-3
  • Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
  • P. Kaiser, Existence theorems in the calculus of variations, Ph.D. Dissertation, Unversity of Michigan, Ann Arbor, Mich., 1973. E. J. McShane, Recent developments in the calculus of variations, Semicentennial addresses, Amer. Math. Soc., 1938, pp. 69-97. E. J. McShane and R. B. Warfield, Jr., On Filippov’s implicit functions lemma, Proc. Amer. Math. Soc. 18 (1967), 41-47; addenda and corrigenda, ibid. 21 (1969), 496-498. MR 34 # 8399; 38 # 6574.
  • Hanno Rund, The Hamilton-Jacobi theory in the calculus of variations: Its role in mathematics and physics, D. Van Nostrand Co., Ltd., London-Toronto, Ont.-New York, 1966. MR 0230189
  • L. Tonelli, Fondamenti di calcolo delle variazioni, Nicola Zanichelli, Bologna, 1921. —, Sugli integrali del calcolo delle variazioni in forma ordinaria, Ann. Scuola Norm. Sup. Pisa 2 (1934), 401-450. L. Turner, The direct method in the calculus of variations, Ph. D. Dissertation, Purdue University, Lafayette, Indiana, 1957.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 49A50
  • Retrieve articles in all journals with MSC: 49A50
Bibliographic Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 223 (1976), 347-365
  • MSC: Primary 49A50
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0425716-3
  • MathSciNet review: 0425716