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We prove that minimizers of the $L^{d}$-norm of the Hessian in the unit ballof $\mathbb{R}^d$ are locally of class $C^{1,\alpha}$. Our findings extendprevious results on Hessian-dependent functionals to the borderline case andresonate with the H\"older regularity theory available for elliptic equationsin double-divergence form.

Improved regularity for a Hessian-dependent functional