Regularity of $p\left(·\right)$-superharmonic functions, the Kellogg property and semiregular boundary points
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, p. 1131-1153

We study various boundary and inner regularity questions for $p\left(·\right)$-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for $p\left(·\right)$-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples.Along the way, we present a removability result for bounded $p\left(·\right)$-harmonic functions and give some new characterizations of ${W}_{0}^{1,p\left(·\right)}$ spaces. We also show that $p\left(·\right)$-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

DOI : https://doi.org/10.1016/j.anihpc.2013.07.012
Classification:  35J67,  31C45,  46E35
Keywords: Comparison principle, Kellogg property, lsc-regularized, Nonlinear potential theory, Nonstandard growth equation, Obstacle problem, $p\left(·\right)$-harmonic, Quasicontinuous, Regular boundary point, Removable singularity, Semiregular point, Sobolev space, Strongly irregular point, $p\left(·\right)$-superharmonic, $p\left(·\right)$-supersolution, Trichotomy, Variable exponent
@article{AIHPC_2014__31_6_1131_0,
author = {Adamowicz, Tomasz and Bj\"orn, Anders and Bj\"orn, Jana},
title = {Regularity of $p(\cdot )$-superharmonic functions, the Kellogg property and semiregular boundary points},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {6},
year = {2014},
pages = {1131-1153},
doi = {10.1016/j.anihpc.2013.07.012},
zbl = {1304.35296},
mrnumber = {3280063},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_6_1131_0}
}

Adamowicz, Tomasz; Björn, Anders; Björn, Jana. Regularity of $p(\cdot )$-superharmonic functions, the Kellogg property and semiregular boundary points. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, pp. 1131-1153. doi : 10.1016/j.anihpc.2013.07.012. http://www.numdam.org/item/AIHPC_2014__31_6_1131_0/

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