Regularity of [formula omitted]-superharmonic functions, the Kellogg property and semiregular boundary points

Abstract

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 W01,p(⋅) spaces. We also show that p(⋅)-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.