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

Under the assumption that the Lagrangian 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.

**[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**

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