Convergence for elliptic equations in periodic perforated domains

Li-Ming Yeh*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let ε denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in ε in L2 space. Also, when ε approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W1,p space for p>1. More precisely, a uniform W1,p estimate in ε for p∈(1, ∞] and a W1,p convergence result for p∈(nn-2,∞] for the elliptic solutions in periodic perforated domains are derived. Here n is the dimension of the space domain. One also notes that the Lp norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in ε.

Original languageEnglish
Pages (from-to)1734-1783
Number of pages50
JournalJournal of Differential Equations
Volume255
Issue number7
DOIs
StatePublished - 1 Oct 2013

Keywords

  • Homogenized elliptic equation
  • Periodic perforated domain

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