Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

The classical problem of the calculus of variations in the autonomous case: Relaxation and Lipschitzianity of solutions


Author: Arrigo Cellina
Journal: Trans. Amer. Math. Soc. 356 (2004), 415-426
MSC (2000): Primary 49N60
DOI: https://doi.org/10.1090/S0002-9947-03-03347-6
Published electronically: June 10, 2003
MathSciNet review: 2020039
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of minimizing

\begin{displaymath}\int _{a}^{b} L(x(t),x^{\prime }(t)) \, dt, \qquad x(a)=A, x(b)=B.\end{displaymath}

Under the assumption that the Lagrangian $L$is continuous and satisfies a growth assumption that does not imply superlinear growth, we provide a result on the relaxation of the functional and show that a solution to the minimum problem is Lipschitzian.


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

  • [A:A:B] L. Ambrosio, O. Ascenzi, and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl. 142 (1989), 301-316. MR 91c:49060
  • [B:M] J. Ball and J. V. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mech. Anal. 90 (1985), 325-388. MR 86k:49002
  • [C:F:M] A. Cellina, A. Ferriero, and E. M. Marchini, Reparametrizations and approximate values of integrals of the calculus of variations, J. Differential Equations, to appear.
  • [C:T:Z] A. Cellina, G. Treu, and S. Zagatti, On the minimum problem for a class of non-coercive functionals, J. Differential Equations 127 (1996), 225-262. MR 97d:49003
  • [Ce] L. Cesari, Optimization, Theory and Applications, Springer-Verlag, New York, 1983. MR 85c:49001
  • [C:V] 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 86h:49020
  • [E:T] I. Ekeland and R. Temam, Analyse convexe et problemes variationnels, Dunod, Paris, 1974. MR 57:3931a
  • [H-U:L] J. B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms. I, Springer-Verlag, Berlin, 1996.
  • [R] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, 1970. MR 43:445
  • [S:V] J. Serrin and D. E. Varberg, A general chain rule for derivatives and the change of variable formula for the Lebesgue integral, Amer. Math. Monthly 76 (1969), 514-520. MR 40:280

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 49N60

Retrieve articles in all journals with MSC (2000): 49N60


Additional Information

Arrigo Cellina
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
Email: cellina@matapp.unimib.it

DOI: https://doi.org/10.1090/S0002-9947-03-03347-6
Keywords: Relaxation, regularity of solutions
Received by editor(s): September 4, 2001
Received by editor(s) in revised form: March 28, 2003
Published electronically: June 10, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society