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Zeitschrift für Analysis und ihre Anwendungen

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Volume 38, Issue 1, 2019, pp. 73–96
DOI: 10.4171/ZAA/1628

Published online: 2019-01-07

Boundary Regularity under Generalized Growth Conditions

Petteri Harjulehto[1] and Peter Hästö[2]

(1) University of Turku, Finland
(2) University of Turku and University of Oulu, Finland

We study the Dirichlet $\phi$-energy integral with Sobolev boundary values. The function $\phi$ has generalized Orlicz growth. Special cases include variable exponent and double phase growths. We show that minimizers are regular at the boundary provided a weak capacity fatness condition is satisfied. This condition is satisfied for instance if the boundary is Lipschitz. The results are new even for Orlicz spaces.

Keywords: Dirichlet energy integral, regular boundary point, minimizer, superminimizer, generalized Orlicz space, Musielak–Orlicz spaces, the weak Harnack inequality, nonstandard growth, variable exponent, double phase

Harjulehto Petteri, Hästö Peter: Boundary Regularity under Generalized Growth Conditions. Z. Anal. Anwend. 38 (2019), 73-96. doi: 10.4171/ZAA/1628