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No additional hypothesis on $F$ is needed here for scalar interior $C^{1,\beta }$ regularity. In the elliptic sense of uniform convexity, the assumptions already give constants $0<\lambda \le M<\infty$ with $\lambda I\le D^{2}F(p)\le MI$ for all $p$. The following argument gives interior gradient regularity for uniformly convex autonomous energies.

The first variation at the minimizer exists because $|DF(p)|\le C(1+|p|)$, and its weak Euler-Lagrange equation isFor a coordinate difference quotient ${\delta }_{h}u$, subtract the translated equation and the original equation. The averaged linearization of a nonlinear divergence-form equation givesThese measurable symmetric matrices satisfy $\lambda I\le A_h\le MI$ independently of $h$. The Caccioppoli inequality and the standard bound $\Vert {\delta }_{h}u{\Vert }_2\le \Vert D_{k}u{\Vert }_2$ bound $D{\delta }_{h}u$ on every smaller interior set. Quoting the Sobolev characterization by bounded difference quotients yields $u\in H_{\mathrm{loc}}^2$.

Now $DF$ is globally Lipschitz continuous, so the Sobolev chain rule and commutation of distributional derivatives allow differentiation of the weak equation. Each $v_k=D_{k}u\in H_{\mathrm{loc}}^1$ satisfiesThe De Giorgi-Nash-Moser theorem states that weak solutions of this scalar divergence-form equation with bounded measurable uniformly elliptic coefficients are locally Hölder continuous. It gives a common $\beta \in (0,1)$ depending only on $n$ and the ellipticity ratio $M/\lambda$, with $D_{k}u\in C_{\mathrm{loc}}^{0,\beta }$ for every $k$. A Sobolev function whose weak gradient is continuous has a $C^1$ representative, as follows by local mollification and integration of its gradient. HenceThe higher-regularity conclusion comes from differentiating an autonomous scalar equation; it does not rely on boundary regularity or a higher derivative such as $D^{3}F$.