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 $\mathbb{R}^{n}$. Let $f≤\left( {x, \xi } \right) $ be a real function defined in $Ω × \mathbb{R}^{n}$ satisfying the growth condition $|{f_{\xi x}}\left( {x, \xi } \right)| \le h\left( x \right)|\xi {{\rm{|}}^{p - 1}}$, for $x∈ Ω $ and $\xi ∈ \mathbb{R}^{n}$ with $|\xi {\rm{|}} \ge {M_0}$ for some $M_{0}≥ 0$, with $h \in L_{{\rm{loc}}}^r\left( \Omega \right) $ 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\left( {x, \xi } \right) $ the so-called natural $p-$growth conditions on the second derivatives ${f_{\xi \xi }}\left( {x, \xi } \right)$; i.e., $\left( {p - 2} \right) - $growth for $|{f_{\xi \xi }}\left( {x, \xi } \right)| $ from above and $\left( {p - 2} \right) - $growth from below for the quadratic form $({f_{\xi \xi }}\left( {x, \xi } \right)\lambda , \lambda {\rm{ }})$; for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer $u \in W_{{\rm{loc}}}^{1, p}\left( \Omega \right) $ of the energy integral $\int_\Omega {f(x, Du\left( x \right)){\mkern 1mu} dx} $.