Improved regularity for a Hessian-dependent functional

Abstract

We prove that minimizers of the L d L^{d} -norm of the Hessian in the unit ball of R d \mathbb {R}^d are locally of class C 1 , α 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.