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

Abstract

Lipschitz, piecewise- C 1 and piecewise affine regularity is proved for AC minimizers of the “affine” integral ∫ a b{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 L loc 1), provided their quotient ϕ/ ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.