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.