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

We consider the problem of minimizing integral(a)(b) 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.