We consider the following functional: \[ J(u)=\int_{\Omega}\{f(Du(x))+g(x,u(x))\}dx, \] where \(\Omega\) is an open bounded subset of \({\mathfrak R}^ n\), u is a vector function from \(\Omega\) into \({\mathfrak R}^ N\), f is a function of class \(C^ 2\) from \({\mathfrak R}^{nN}\) into \({\mathfrak R}\), g is defined on \({\mathfrak R}^{n+N}\) and Du, the gradient of u, is defined on \(\Omega\) and takes values in \({\mathfrak R}^{nN}.\)
A function u belonging to \(W^{1,m}(\Omega;{\mathfrak R}^ N)\) and satisfying some boundary conditions (here unspecified), is a minimiser of J if, for every function \(\phi\) in \(W_ 0^{1,m}(\Omega;{\mathfrak R}^ N)\), with compact support in \(\Omega\), we have \(J(u)\leq J(u+\phi)\). Here, we suppose that the existence of minimisers for J is proved and we are only concerned with the regularity of such minimisers. More precisely, we show that, if f is closed enough at infinity to the function \(P\to | P|^ m\), with \(m\geq 2\), minimisers of J belong to \(W_{loc}^{1,\infty}(\Omega;{\mathfrak R}^ N)\). We thus extend previous results due to M. Giaquinta and G. Modica [Manuscr. Math. 57, 55-99 (1986; Zbl 0607.49003)] and M. Chipot and L. C. Evans [Proc. R. Soc. Edinb., Sect. A 102, 291-303 (1986; Zbl 0602.49029)] in the case where g is different from the zero function. As Chipot-Evans and Giaquinta-Modica we do not make any convexity or quasiconvexity assumption of f. At the end of this paper, we give some examples of polyconvex functions f which satisfy sufficient growth conditions to obtain regularity results.