 Interior gradient regularity for uniformly convex autonomous energies

For scalar minimizers of $\int F(Du)$ with $F\in C^2({\mathbb{R}}^n)$ and globally bounded uniformly positive Hessian matrix, no higher differentiability of $F$ is needed for interior gradient Hölder continuity. The weak Euler-Lagrange equation is $\mathrm{div}DF(Du)=0$. Difference quotients solve uniformly elliptic equations with coefficient matrices obtained by averaging $D^{2}F$ along gradient segments. The Caccioppoli inequality and difference quotient characterization give $u\in H_{\mathrm{loc}}^2$. The Sobolev chain rule then gives $\mathrm{div}(D^{2}F(Du)D(D_{k}u))=0$. The De Giorgi-Nash-Moser theorem makes each derivative Hölder continuous, with exponent depending only on dimension and the ellipticity ratio. This scalar conclusion should not be assumed for vector-valued elliptic systems.