The Approximation of Higher-Order Integrals of the Calculus of Variations and the Lavrentiev Phenomenon

x (ν) (a) = x (a), x (ν) (b) = x (b), provided that, for every i in {1, . . . ,m}, Liψi is continuous in a neighborhood of x, Li is convex in its second variable, and ψi 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 ∫ b a L(x (ν), x(ν+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(ν)(a) = A(ν), x(ν)(b) = B(ν). Furthermore, we prove the following necessary condition: an action functional with Lagrangian of the form ∑m i=1 Li(x (ν), x)ψi(t, x, x ′, . . . , x(ν)), with ν ≥ 0, exhibiting the Lavrentiev phenomenon takes the value +∞ in any neighborhood of a minimizer.