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We consider the problem of minimizing b a L(x(t), x'(t))dt, x(a) = A,x(b) = B. 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.

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