On the regularity of minimizers for scalar integral functionals with $(p,q)$-growth

Abstract

We revisit the question of regularity for minimizers of scalar autonomous integral functionals with so-called $\left(p,q\right)$-growth. In particular, we establish Lipschitz regularity under the condition $\frac{q}{p}<1+\frac{2}{n-1}$ for $n\ge 3$, improving a classical result due to Marcellini (J. Differential Equations 90:1 (1991), 1–30).