The approximation of higher-order integrals of the calculus of variations and the Lavrentiev phenomenon

We prove the following approximation theorem: given a function x : [a, b] --> R-N in the Sobolev space W-nu+1,W-1, nu >= 1, and epsilon > 0, there exists a function x(epsilon) in W-nu+ 1,W-infinity such that integral(a)(b) Sigma(i=1)(m) L-i(x(epsilon)((nu)), x(epsilon)((nu+1))psi(i)(t, x(epsilon), x'(epsilon),...,x(epsilon)((nu))) < integral(a)(b) Sigma(i=1)(m) L-i(x((nu)) ,x((nu+1))psi(i)(t, x,x',...,x((nu)) + epsilon, x(epsilon)(a) = x(a), x(epsilon)(b) = x(b), x'(epsilon)(a) = x'(a), x'(epsilon)(b) = x'(b), 3 vertical dots x(epsilon)((nu))(a) = x((nu))(a), x(epsilon)((nu))(b) = x((nu))(b), provided that, for every i in {1,..., m}, L(i)psi(i) is continuous in a neighborhood of x, L-i is convex in its second variable, and psi(i) evaluated along x has positive sign. We discuss the optimality of our assumptions comparing them with an example of Sarychev [J. Dynam. Control Systems, 3 ( 1997), pp. 565 - 588]. As a consequence, we obtain the nonoccurrence of the Lavrentiev phenomenon. In particular, the integral functional integral(a)(b) L(x((nu)), x((nu+1))) does not exhibit the Lavrentiev phenomenon for any given boundary values x(a) = A, x(b) = B, x'(a) = A', x'(b) = B',..., x((nu))(a) = A((nu)), x((nu))(b) = B-(nu). Furthermore, we prove the following necessary condition: an action functional with Lagrangian of the form Sigma(i=1)(m) L-i(x((nu)), x((nu+1)))psi(i)(t, x, x',..., x((nu))), with nu >= 0, exhibiting the Lavrentiev phenomenon takes the value +infinity in any neighborhood of a minimizer.