Generic Uniqueness of a Minimal Solution for Variational Problems on a Torus

where a and b are arbitrary real numbers satisfying a < b, x ∈ W1,1(a,b) and f belongs to a space of functions described below. By an appropriate choice of representatives, W1,1(a,b) can be identified with the set of absolutely continuous functions x : [a,b] → R1, and henceforth we will assume that this has been done. Denote by M the set of integrands f = f (t,x, p) : R3 → R1 which satisfy the following assumptions: (A1) f ∈ C3 and f (t,x, p) has period 1 in t,x; (A2) δ f ≤ fpp(t,x, p) ≤ δ−1 f for every (t,x, p) ∈ R3; (A3) | fxp|+ | ftp| ≤ c f (1+ |p|), | fxx|+ | fxt| ≤ c f (1+ p2), with some constants δ f ∈ (0,1), c f > 0. Clearly, these assumptions imply that