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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
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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.


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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