Abstract

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $$-growth with exponents $p \le q < \infty $ and show for the scalar case that locally bounded local minimizers are of class $C^{1, \mu }$. Note that to our knowledge the only $C^{1,\mu }$-results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2