Lipschitz regularity for scalar minimizers of autonomous simple integrals

AbstractWe prove Lipschitz regularity for a minimizer of the integral ∫abL(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