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arXiv · 1302.0233

Regularity of $p(\cdot)$-superharmonic functions, the Kellogg property and semiregular boundary points

Tomasz Adamowicz·Anders Björn·Jana Björn
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Abstract

We study various boundary and inner regularity questions for $p(\cdot)$-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for $p(\cdot)$-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 $p(\cdot)$-harmonic functions and give some new characterizations of $W^{1, p(\cdot)}_0$ spaces. We also show that $p(\cdot)$-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

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Tomasz Adamowicz, Anders Björn, Jana Björn. 2013-02-01. Regularity of $p(\cdot)$-superharmonic functions, the Kellogg property and semiregular boundary points. https://doi.org/10.1016/j.anihpc.2013.07.012

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