Continuity of solutions to n-harmonic equations |
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Authors: | Renjin Jiang Pekka Koskela Dachun Yang |
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Institution: | 1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China 2. Department of Mathematics and Statistics, University of Jyv?skyl?, P.O. Box 35 (MaD), 40014, Jyv?skyl?, Finland
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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 ${\epsilon=0}$ . |
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