Lipschitz and piecewise-C1 regularity for scalar minimizers of affine simple integrals

AbstractLipschitz, piecewise-C1 and piecewise affine regularity is proved for AC minimizers of the “affine” integral ∫ab{ρ(x)h(x′)+ϕ(x)}dt, under general hypotheses on ρ:R→[1,+∞), ϕ:R→R, and h:R→[0,+∞] with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ϕ(·) are lsc and may be both locally unbounded (e.g., not in Lloc1), provided their quotient ϕ/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely