= Interior gradient regularity for uniformly convex autonomous energies
{title2=$\lambda I\leq D^2F\leq MI\Longrightarrow u\in C^{1,\beta}_{\mathrm{loc}}$}
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 $\operatorname{div}DF(Du)=0$. Difference quotients solve uniformly elliptic equations with coefficient matrices obtained by averaging $D^2F$ along gradient segments. The <Caccioppoli inequality> and difference quotient characterization give $u\in H^2_{\mathrm{loc}}$. The <Sobolev chain rule> then gives $\operatorname{div}(D^2F(Du)D(D_ku))=0$. 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.
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