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), p. 1131-1153 / Harvested from Numdam
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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.

Publié le : 2014-01-01
DOI : https://doi.org/10.1016/j.anihpc.2013.07.012
Classification:  35J67,  31C45,  46E35 
@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},
     volume = {31},
     year = {2014},
     pages = {1131-1153},
     doi = {10.1016/j.anihpc.2013.07.012},
     mrnumber = {3280063},
     zbl = {1304.35296},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 31 (2014) pp. 1131-1153. doi : 10.1016/j.anihpc.2013.07.012. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_6_1131_0/

[1] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213 -259 | MR 1930392 | Zbl 1038.76058

[2] E. Acerbi, G. Mingione, Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math. 584 (2005), 117 -148 | MR 2155087 | Zbl 1093.76003

[3] T. Adamowicz, P. Hästö, Mappings of finite distortion and PDE with nonstandard growth, Int. Math. Res. Not. 2010 (2010), 1940 -1965 | MR 2646346 | Zbl 1206.35134

[4] T. Adamowicz, P. Hästö, Harnack's inequality and the strong p(·)-Laplacian, J. Differ. Equ. 250 (2011), 1631 -1649 | MR 2737220 | Zbl 1205.35084

[5] Yu.A. Alkhutov, O.V. Krasheninnikova, Continuity at boundary points of solutions of quasilinear elliptic equations with a nonstandard growth condition, Izv. Ross. Akad. Nauk Ser. Mat. 68 no. 6 (2004), 3 -60 , Izv. Math. 68 (2004), 1063 -1117 | MR 2108520 | Zbl 1167.35385

[6] D.H. Armitage, S.J. Gardiner, Classical Potential Theory, Springer, London (2001) | MR 1801253

[7] A. Björn, A regularity classification of boundary points for p-harmonic functions and quasiminimizers, J. Math. Anal. Appl. 338 (2008), 39 -47 | MR 2386397 | Zbl 1143.31006

[8] A. Björn, J. Björn, Boundary regularity for p-harmonic functions and solutions of the obstacle problem, J. Math. Soc. Jpn. 58 (2006), 1211 -1232 | MR 2276190 | Zbl 1211.35109

[9] A. Björn, J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts Math. vol. 17 , European Math. Soc., Zürich (2011) | MR 2867756 | Zbl 1231.31001

[10] A. Björn, J. Björn, U. Gianazza, M. Parviainen, Boundary regularity for degenerate and singular parabolic equations, in preparation.

[11] A. Björn, J. Björn, N. Shanmugalingam, The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173 -203 | MR 1971145 | Zbl 1018.31004

[12] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 no. 4 (2006), 1383 -1406 | MR 2246061 | Zbl 1102.49010

[13] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lect. Notes Math. vol. 2017 , Springer, Berlin, Heidelberg (2011) | MR 2790542 | Zbl 1222.46002

[14] X.-L. Fan, Global C 1,α regularity for variable exponent elliptic equations in divergence form, J. Differ. Equ. 235 (2007), 397 -417 | Zbl 1143.35040

[15] X.-L. Fan, Y.Z. Zhao, Q.-H. Zhang, A strong maximum principle for p(x)-Laplace equations, Chin. Ann. Math., Ser. A 24 (2003), 495 -500 , Chin. J. Contemp. Math. 24 (2003), 277 -282

[16] P. Harjulehto, P. Hästö, M. Koskenoja, T. Lukkari, N. Marola, An obstacle problem and superharmonic functions with nonstandard growth, Nonlinear Anal. 67 (2007), 3424 -3440 | MR 2350898 | Zbl 1130.31004

[17] P. Harjulehto, P. Hästö, V. Latvala, O. Toivanen, The strong minimum principle for quasisuperminimizers of non-standard growth, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28 (2011), 731 -742 | Numdam | MR 2838399 | Zbl 1251.49028

[18] P. Harjulehto, P. Hästö, Ú.V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), 4551 -4574 | MR 2639204 | Zbl 1188.35072

[19] P. Harjulehto, J. Kinnunen, T. Lukkari, Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. Value Probl. 2007 (2007) | MR 2291928 | Zbl 1161.35020

[20] P. Harjulehto, V. Latvala, Fine topology of variable exponent energy superminimizers, Ann. Acad. Sci. Fenn. Math. 33 (2008), 491 -510 | MR 2431377 | Zbl 1152.31005

[21] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York (2001) | MR 1800917 | Zbl 0985.46008

[22] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover, Mineola, NY (2006) | MR 2305115 | Zbl 0776.31007

[23] E. Henriques, Regularity for the porous medium equation with variable exponent: the singular case, J. Differ. Equ. 244 (2008), 2578 -2601 | MR 2414406 | Zbl 1157.35020

[24] T. Kilpeläinen, A remark on the uniqueness of quasi continuous functions, Ann. Acad. Sci. Fenn. Math. 23 no. 1 (1998), 261 -262 | MR 1601887 | Zbl 0919.31006

[25] V. Latvala, T. Lukkari, O. Toivanen, The fundamental convergence theorem for p(·)-superharmonic functions, Potential Anal. 35 (2011), 329 -351 | MR 2846295 | Zbl 1228.31005

[26] H. Lebesgue, Sur des cas d'impossibilité du problème de Dirichlet ordinaire, in Vie de la société, Bull. Soc. Math. Fr. 41 (1913), 1 -62

[27] J. Lukeš, J. Malý, On the boundary behaviour of the Perron generalized solution, Math. Ann. 257 (1981), 355 -366 | MR 637957 | Zbl 0461.31003

[28] T. Lukkari, Singular solutions of elliptic equations with nonstandard growth, Math. Nachr. 282 (2009), 1770 -1787 | MR 2588835 | Zbl 1180.35253

[29] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lect. Notes Math. vol. 1748 , Springer, Berlin (2000) | MR 1810360 | Zbl 0968.76531

[30] S. Zaremba, Sur le principe de Dirichlet, Acta Math. 34 (1911), 293 -316 | JFM 42.0393.01 | MR 1555069