Existence theorems for parametric problems in the calculus of variations and approximation

Author:
Robert M. Goor

Journal:
Trans. Amer. Math. Soc. **223** (1976), 347-365

MSC:
Primary 49A50

MathSciNet review:
0425716

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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.**[2]**Lamberto Cesari,*Closure theorems for orientor fields and weak convergence*, Arch. Rational Mech. Anal.**55**(1974), 332–356. MR**0350589****[3]**Lamberto Cesari,*Rectifiable curves and the Weierstrass integral*, Amer. Math. Monthly**65**(1958), 485–500. MR**0131504****[4]**Lamberto Cesari,*Seminormality and upper semicontinuity in optimal control*, J. Optimization Theory Appl.**6**(1970), 114–137. MR**0270248****[5]**-,*Surface area*, Ann. of Math. Studies, no. 35, Princeton Univ. Press, Princeton, N.J., 1956. MR**17**, 596.**[6]**George M. Ewing,*Calculus of variations with applications*, W. W. Norton & Co. Inc., New York, 1969. MR**0242032****[7]**Robert M. Goor,*Gradient approximation of vector fields*, J. Approximation Theory**12**(1974), 385–395. MR**0355440****[8]**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493****[9]**P. Kaiser,*Existence theorems in the calculus of variations*, Ph.D. Dissertation, Unversity of Michigan, Ann Arbor, Mich., 1973.**[10]**E. J. McShane,*Recent developments in the calculus of variations*, Semicentennial addresses, Amer. Math. Soc., 1938, pp. 69-97.**[11]**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.**[12]**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****[13]**L. Tonelli,*Fondamenti di calcolo delle variazioni*, Nicola Zanichelli, Bologna, 1921.**[14]**-,*Sugli integrali del calcolo delle variazioni in forma ordinaria*, Ann. Scuola Norm. Sup. Pisa**2**(1934), 401-450.**[15]**L. Turner,*The direct method in the calculus of variations*, Ph. D. Dissertation, Purdue University, Lafayette, Indiana, 1957.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
49A50

Retrieve articles in all journals with MSC: 49A50

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0425716-3

Keywords:
Parametric problem,
calculus of variations,
orientor field,
growth condition,
Fréchet curve,
parametric problems of optimal control

Article copyright:
© Copyright 1976
American Mathematical Society