Regularity for scalar integrals without structure conditions

Integrals of the Calculus of Variations with p, q-growth may have not smooth minimizers, not even bounded, for general p, q exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f (x, ξ) with dependence on the modulus of the gradient, i.e. f(x , ξ) = g (x,|ξ|). Without imposing structure conditions, we prove that if q p is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.