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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
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

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

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