Bloch estimates in non-doubling generalized Orlicz spaces

Abstract

We study minimizers of non-autonomous functionals $ \begin{align*} \inf\limits_u \int_\Omega \varphi(x,|\nabla u|) \, dx \end{align*} $ when $ \varphi $ has generalized Orlicz growth. We consider the case where the upper growth rate of $ \varphi $ is unbounded and prove the Harnack inequality for minimizers. Our technique is based on "truncating" the function $ \varphi $ to approximate the minimizer and Harnack estimates with uniform constants via a Bloch estimate for the approximating minimizers.

Keywords

• non-doubling
• harnack's inequality
• generalized orlicz space
• musielak–orlicz spaces
• nonstandard growth
• variable exponent
• double phase