Quantified Legendreness and the regularity of minima

We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and $2d$-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples - in particular, we improve in an essentially optimal fashion Marcellini's original results \cite{ma1}.