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

Cellina, Arrigo

doi:10.1090/S0002-9947-03-03347-6

The classical problem of the calculus of variations in the autonomous case: Relaxation and Lipschitzianity of solutions
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Trans. Amer. Math. Soc. 356 (2004), 415-426 Request permission

Abstract:

We consider the problem of minimizing \begin{equation*}\int _{a}^{b} L(x(t),x^{\prime }(t)) dt, \qquad x(a)=A, x(b)=B.\end{equation*} 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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