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

Lipschitz, piecewise-C1 and piecewise affine regularity is proved for AC minimizers of the “affine” integral b a {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 L1 loc), provided their quotient ϕ/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.  2004 Elsevier Inc. All rights reserved