Lipschitz regularity for scalar minimizers of autonomous simple integrals

We prove Lipschitz regularity for a minimizer of the integral , defined on the class of the AC functions having x(a)=A and x(b)=B. The Lagrangian may have L(s,) nonconvex (except at ξ=0), while may be non-lsc, measurability sufficing for ξ≠0 provided, e.g., L^**() is lsc at (s,0) s. The essential hypothesis (to yield Lipschitz minimizers) turns out to be local boundedness of the quotient φ/ρ() (and not of L^**() itself, as usual), where φ(s)+ρ(s)h(ξ) approximates the bipolar L^**(s,ξ) in an adequate sense. Moreover, an example of infinite Lavrentiev gap with a scalar 1-dim autonomous (but locally unbounded) lsc Lagrangian is presented.