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Manuscripta Mathematica

, Volume 139, Issue 1–2, pp 237–248 | Cite as

Continuity of solutions to n-harmonic equations

  • Renjin Jiang
  • Pekka Koskela
  • Dachun Yang
Article

Abstract

In this paper, we study the nonhomogeneous n-harmonic equation
$$-{\rm div}\,(|{\nabla} u|^{n-2}{\nabla} u)=f$$
in domains \({\Omega\subset {\mathbb {R}^n}}\) (n ≥ 2), where \({f\in W^{-1,\frac{n}{n-1}}(\Omega)}\). We derive a sharp condition to guarantee the continuity of solutions u. In particular, we show that when n ≥ 3, the condition that, for some \({\epsilon >0 ,}\) f belongs to
$${\mathfrak{L}}({\rm log}\,{\mathfrak{L}})^{n-1}({\rm log}\,{\rm log}\,{\mathfrak{L}})^{n-2}\cdots({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2}({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2+\epsilon}(\Omega)$$
is sufficient for continuity of u, but not for \({\epsilon=0}\).

Mathematics Subject Classification (2010)

Primary 35J60 Secondary 31B35 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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