Local Lipschitz continuity of minimizers with mild assumptions on the x-dependence

We are interested in the regularity of local minimizers of energy integrals of the Calculus of Variations. Precisely, let be an open subset of Rn. Let f (x, ) be a real function defined in × Rn satisfying the growth condition fξx (x, ) h (x) ||p−1, for x 2 and 2 Rn with || M0 for some M0 0, with h 2 Lrloc () for some r > n. This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to f (x, ) the so-called natural p−growth conditions on the second derivatives fξξ (x, ); i.e., (p − 2)−growth for fξξ (x, ) from above and (p − 2)−growth from below for the quadratic form (fξξ (x, ) , ); for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer u 2 W1,ploc () of the energy integral R f (x,Du (x)) dx .