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

We consider the problem of minimizing ∫ a b L ( x ( t ) , x ′ ( t ) ) d t , x ( a ) = A , x ( b ) = B . \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 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