Improved regularity for a Hessian-dependent functional

We prove that minimizers of the $L^d$-norm of the Hessian in the unit ball of ${\mathbb{R}}^d$ are locally of class $C^{1,\alpha }$. Our findings extend previous results on Hessian-dependent functionals to the borderline case and resonate with the Hölder regularity theory available for elliptic equations in double-divergence form.