Lipschitz regularity for scalar minimizers of autonomous simple integrals

We prove Lipschitz regularity for a minimizer of the integral b a L(x, x )dt, defined on the class of the AC functions x : [a, b] → R having x(a) = A and x(b) = B. The Lagrangian L:R × R→ [0,+∞] may have L(s, ·) nonconvex (except at ξ = 0), while L(·, ξ) 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.  2004 Elsevier Inc. All rights reserved