Bloch estimates in non-doubling generalized Orlicz spaces

Petteri Harjulehto; Peter Hästö; Jonne Juusti; Petteri Harjulehto; Peter Hästö; Jonne Juusti

doi:10.3934/mine.2023052

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-21. doi: 10.3934/mine.2023052

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Bloch estimates in non-doubling generalized Orlicz spaces

1.
Department of Mathematics and Statistics, FI-00014 University of Helsinki, Finland
2.
Department of Mathematics and Statistics, FI-20014 University of Turku, Finland

Received: 01 July 2022 Revised: 09 August 2022 Accepted: 19 September 2022 Published: 23 September 2022

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.
- non-doubling,
- Harnack's inequality,
- generalized Orlicz space,
- Musielak–Orlicz spaces,
- nonstandard growth,
- variable exponent,
- double phase
Citation: Petteri Harjulehto, Peter Hästö, Jonne Juusti. Bloch estimates in non-doubling generalized Orlicz spaces[J]. Mathematics in Engineering, 2023, 5(3): 1-21. doi: 10.3934/mine.2023052

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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.

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