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 Ω be an open subset of Rn. Let f (x, ξ) be a real function defined in Ω × Rnsatisfying the growth condition |fξx(x, ξ)| ≤ h (x) |ξ|p−1, for x ∈ Ω and ξ ∈ Rnwith |ξ| ≥ M0for some M0≥ 0, with h ∈ Lrloc(Ω) 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, ξ) the so-called natural p−growth conditions on the second derivatives fξξ(x, ξ); i.e., (p − 2) −growth for |fξξ(x, ξ)| from above and (p − 2) −growth from below for the quadratic form (fξξ(x, ξ) λ, λ); for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer u ∈ Wloc1,p(Ω) of the energy integral fΩf (x, Du (x)) dx .