Regularity of p(·)-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
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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.
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(·)-harmonic, Quasicontinuous, Regular boundary point, Removable singularity, Semiregular point, Sobolev space, Strongly irregular point, p(·)-superharmonic, p(·)-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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