Bloch estimates in non-doubling generalized Orlicz spaces

TL;DR

This paper establishes Harnack inequalities for minimizers of non-autonomous functionals with generalized Orlicz growth, even when the growth rate is unbounded, using a novel truncation and Bloch estimate technique.

Contribution

It introduces a new approach employing truncation and Bloch estimates to prove Harnack inequalities in non-doubling generalized Orlicz spaces.

Findings

Proved Harnack inequality for minimizers with unbounded growth.

Developed a truncation method to approximate minimizers.

Established uniform constants in Harnack estimates.

Abstract

We study minimizers of non-autonomous functionals \begin{align*} \inf_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.