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

Adamowicz, Tomasz; Björn, Anders; Björn, Jana

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1131-1153.

We study various boundary and inner regularity questions for $p(·)$-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for $p(·)$-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 $p(·)$-harmonic functions and give some new characterizations of $W_0^{1,p(·)}$ spaces. We also show that $p(·)$-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

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