Abstract
We study regularity properties of weak solutions to elliptic equations involving variable growth exponents. We prove the sufficiency of a Wiener type criterion for the regularity of boundary points. This criterion is formulated in terms of the natural capacity involving the variable growth exponent. We also prove the Hölder continuity of weak solutions up to the boundary in domains with uniformly fat complements, provided that the boundary values are Hölder continuous.
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Acerbi E., Mingione G.: Regularity results for a class of functionals with nonstandard growth. Arch. Rational Mech. Anal. 156, 121–140 (2001)
Acerbi E., Mingione G.: Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math. 584, 117–148 (2005)
Alkhutov Y.A.: The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition. Differ. Equ. 33, 1653–1663 (1997)
Alkhutov Y.A., Krasheninnikova O.V.: Continuity at boundary points of solutions of quasilinear elliptic equations with a nonstandard growth condition. Izv. Ross. Akad. Nauk Ser. Mat. 68, 3–60 (2004)
Diening L.: Maximal function on generalized Lebesgue spaces L p(·). Math. Inequal. Appl. 7, 245–253 (2004)
Fan X., Zhao D.: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 36, 295–318 (1999)
Gariepy R., Ziemer W.P.: A regularity condition at the boundary for solutions of quasilinear elliptic equations. Arch. Rational Mech. Anal. 67, 25–39 (1977)
Harjulehto, P., Kinnunen, J., Lukkari, T.: Unbounded supersolutions of nonlinear equations with nonstandard growth, 20 pp. Bound. Value Probl. vol. 2007, Article ID 48348 (2007). http://www.hindawi.com/GetArticle.aspx? doi:10.1155/2007/48348
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Dover Publications Inc., Mineola, NY (2006). ISBN 0-486-45050-3. Unabridged republication of the 1993 original
Kilpeläinen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)
Kováčik O., Rákosník J.: On spaces L p(x) and W 1,p(x). Czechoslovak Math. J. 41, 592–618 (1991)
Ladyzhenskaya, O.A., Ural′tseva, N.N.: Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York (1968)
Lukkari, T., Maeda, F.-Y., Marola, N.: Wolff potential estimates for elliptic equations with nonstandard growth and applications. Forum Math. (2010). doi:10.1515/FORUM.2010.057
Malý J.: Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points. Comment. Math. Univ. Carolin. 37, 23–42 (1996)
Malý, J., Ziemer, W.P.: Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51. American Mathematical Society, Providence (1997). ISBN 0-8218-0335-2
Maz’ya, V.G.: The continuity at a boundary point of the solutions of quasi-linear elliptic equations, pp. 42–55. Vestnik Leningrad. Univ. vol. 25 (1970). English translation: Vestnik Leningrad, pp. 225–242. Univ. Math. vol. 3 (1976)
Samko, S.: Denseness of \({C^\infty_0({\bf R}^N)}\) in the generalized Sobolev spaces \({W^{M,P(X)}({\bf R}^N)}\). In: Direct and inverse problems of mathematical physics (Newark, DE, 1997), Int. Soc. Anal. Appl. Comput. vol. 5, pp. 333–342. Kluwer, Dordrecht (2000)
Zhikov V.V.: On Lavrentiev’s phenomenon. Russian J. Math. Phys. 3, 249–269 (1995)
Zhikov V.V.: On some variational problems. Russian J. Math. Phys. 5, 105–116 (1997)
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Lukkari, T. Boundary continuity of solutions to elliptic equations with nonstandard growth. manuscripta math. 132, 463–482 (2010). https://doi.org/10.1007/s00229-010-0355-3
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DOI: https://doi.org/10.1007/s00229-010-0355-3