Lp gradient estimate for elliptic equations with high-contrast conductivities in Rn

Li-Ming Yeh*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Uniform estimate for the solutions of elliptic equations with high-contrast conductivities in Rn is concerned. The space domain consists of a periodic connected sub-region and a periodic disconnected matrix block subset. The elliptic equations have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Suppose ε∈(0, 1] is the diameter of each matrix block and ω2∈(0, 1] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W1,p norm of the elliptic solutions in the connected sub-region is bounded uniformly in ε, ω when ε≤ω, the Lp norm of the elliptic solutions in the whole space is bounded uniformly in ε, ω the W1,p norm of the elliptic solutions in perforated domains is bounded uniformly in ε. However, the Lp norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in ε, ω.

Original languageEnglish
Pages (from-to)925-966
Number of pages42
JournalJournal of Differential Equations
Volume261
Issue number2
DOIs
StatePublished - 15 Jul 2016

Keywords

  • Duality argument
  • Embedding theory
  • High-contrast conductivity
  • Potentials

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