Abstract
We study the balayage related to the supersolutions of the variable exponent p(·)-Laplace equation. We prove the fundamental convergence theorem for the balayage and apply it for proving the Kellogg property, boundary regularity results for the balayage, and a removability theorem for p(·)-solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acerbi, E., Mingione, G.: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)
Alkhutov, Yu.A.: The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition. Diff. Equ. 33(12), 1653–1663 (1997)
Alkhutov, Yu.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(6), 3–60 (2004)
Björn, A., Björn, J., Parviainen, M.: Lebesgue points and the fundamental convergence theorem for superharmonic functions. Rev. Mat. Iberoam. 26(1), 147–174 (2010)
Björn, A., Björn, J., Mäkäläinen, T., Parviainen, M.: Nonlinear balayage on metric spaces. Nonlinear Anal. 71(5–6), 2153–2171 (2009)
Björn, A., Björn, J., Shanmugalingam, N.: Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces. Houston J. Math. 34(4), 1197–1211 (2008)
Brelot, M.: Minorantes sousharmoniques, extrémales et capaciles. J. Math. Pures Appl. 207(9), 836–839 (1945)
Cartan, H.: Theorie du potential newtonian: energie, capacite, suites de potentiels. Bull. Soc. Math. France 73, 74–106 (1945)
Diening, L.: Maximal function on generalized Lebesgue spaces L p(·). Math. Inequal. Appl. 7(2), 245–253 (2004)
Doob, J.L.: Classical potential theory and its probabilistic counterpart. In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 262. Springer, New York (1984)
Fan, X., Zhao, D.: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 36(3), 295–318 (1999)
Harjulehto, P.: Variable exponent Sobolev spaces with zero boundary values. Math. Bohem. 132(2), 125–136 (2007)
Harjulehto, P., Hästö, P., Koskenoja, M.: Properties of capacities in variable exponent Sobolev spaces. J. Anal. Appl. 5(2), 71–92 (2007)
Harjulehto, P., Hästö, P., Koskenoja, M., Lukkari, T., Marola, N.: An obstacle problem and superharmonic functions with nonstandard growth. Nonlinear Anal. 67(12), 3424–3440 (2007)
Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: Sobolev capacity on the space W 1,p(·)(ℝn). J. Funct. Spaces Appl. 1, 17–33 (2003)
Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. Potential Anal. 25(3), 205–222 (2006)
Harjulehto, P., Hästö, P., Latvala, V.: Sobolev embeddings in metric measure spaces with variable dimension. Math. Z. 254(3), 591–609 (2006)
Harjulehto, P., Kinnunen, J., Lukkari, T.: Unbounded supersolutions of nonlinear equations with nonstandard growth. Bound. Value Probl. Article ID 48348, vol. 2007, 20 pp. (2007). Available at http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2007/48348
Harjulehto, P., Latvala, V.: Fine topology of variable exponent energy superminimizers. Ann. Acad. Sci. Fenn. Math. 33(2), 491–510 (2008)
Hästö, P.: On the density of continuous functions in variable exponent Sobolev space. Rev. Mat. Iberoamer. 23(1), 215–237 (2007)
Heinonen, J., Kilpeläinen, T.: A-superharmonic functions and supersolutions of degenerate elliptic equations. Ark. Mat. 26(1), 87–105 (1988)
Heinonen, J., Kilpeläinen, T.: Polar sets for supersolutions of degenerate elliptic equations. Math. Scand. 63, 136–159 (1988)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover, Mineola (2006). Unabridged republication of the 1993 original
Kilpeläinen, T., Zhong, X.: Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Am. Math. Soc. 130(6), 1681–1688. Electronic (2002)
Kilpeläinen, T.: A remark on the uniqueness of quasi continuous functions. Ann. Acad. Sci. Fenn. Math. 23(1), 261–262 (1998)
Kinnunen, J., Martio, O.: Nonlinear potential theory on metric spaces. Ill. Math. J. 46, 857–883 (2002)
Kinnunen, J., Shanmugalingam, N.: Polar sets on metric spaces. Trans. Am. Math. Soc. 358, 11–37 (2006)
Kováčik, O., Rákosník, J.: On spaces L p(x) and W 1,p(x). Czech. Math. J. 41(116), 592–618 (1991)
Lindqvist, P.: On the definition and properties of p-superharmonic functions. J. Reine Angew. Math. 365, 67–79 (1986)
Lukkari, T.: Elliptic equations with nonstandard growth involving measures. Hiroshima Math. J. 38(1), 155–176 (2008)
Lukkari, T.: Singular solutions of elliptic equations with nonstandard growth. Math. Nachr. 282(12), 1770–1787 (2009)
Samko, S.: Denseness of \(C\sp \infty\sb 0(\bold R\sp N)\) in the generalized Sobolev spaces \(W\sp {M,P(X)}(\bold R\sp N)\). 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. Russ. J. Math. Phys. 3(2), 249–269 (1995)
Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5(1), 105–116 (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Latvala, V., Lukkari, T. & Toivanen, O. The Fundamental Convergence Theorem for p(·)-Superharmonic Functions. Potential Anal 35, 329–351 (2011). https://doi.org/10.1007/s11118-010-9215-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-010-9215-8
Keywords
- Non-standard growth
- p(·)-Laplacian
- Comparison principle
- Fundamental convergence theorem
- Boundary regularity
- Removability