Bloch estimates in non-doubling generalized Orlicz spaces

Abstract

We study minimizers of non-autonomous functionals inf u ˆΩ ϕ(x, |∇u|) dx when ϕ has generalized Orlicz growth. We consider the case where the upper growth rate of ϕ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function ϕ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.