Let \(f: \mathbb R^{Nn} \to \mathbb R\) be a \(C^2\) function and consider the problem of minimizing the functional \[ I(u) = \int_\Omega f(Du) dx \] over a class of functions \(u\) defined on \(\Omega\), a domain in \(\mathbb R^n\), with values in \(\mathbb R^N\). If \(f\) is similar to \(f(z) = |z|^p\) for some \(p \geq 2\), the regularity theory for such problems is well understood, and any minimizing function with gradient in \(L^p\) actually has gradient in \(L^{p +\delta}\) for some positive \(\delta\). In this work, the authors consider a more general structure for \(f\). They assume that there are constants \(L\geq 0\), \(\nu >0\), \(p \geq 2\), and \(q \in (p,p+ 2\min\{1,p/n\})\) such that \[ \begin{gathered} |z|^p \leq f(z) \leq L(1+|z|^q),\;|D^2f(z)|\leq L(1+|z|^{q-2}), \\ \frac {\partial^2f}{\partial z^\alpha_i \partial z ^\beta_j} \lambda^\alpha_i \lambda^\beta_j \geq \nu|z|^{p-2}|\lambda|^2 \end{gathered} \] for all \(z\) and \(\lambda\) in \(\mathbb R^{Nn}\). By employing a clever bootstrap argument, they show that any minimizer with gradient in \(L^p\) actually has gradient in \(L ^q\). Similar results are proved for the functional \[ J(u) = \int_\Omega f(Du) +a(x)u dx \] under various sets of appropriate regularity assumptions on \(a\). In all cases, the authors estimate the integral of \(|Du|^q\) by a function that grows like a power of the integral of \(f(Du)\).