An elliptic regularity result for a composite medium with "touching" fibers of circular cross-section

Eric Bonnetier, Michael Vogelius

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76 Scopus citations

Abstract

In this paper we consider the elliptic equation ▽ · α▽u = 0 in a two dimensional domain Ω, which contains a finite number of circular inhomogeneities (cross-sections of fibers). The coefficient, a, takes two constant values, one in all the inhomogeneities and one in the part of Ω which lies outside the inhomogeneities. A number of the inhomogeneities may possibly touch, but in spite of this we prove that any variational solution u (with sufficiently smooth boundary data) is in W1,∞. For this very interesting, particular type of coefficient, our result improves a classical regularity result due to DeGiorgi and Nash, which asserts that the solution is in the Hölder class Cγ for some positive exponent γ.

Original languageEnglish (US)
Pages (from-to)651-677
Number of pages27
JournalSIAM Journal on Mathematical Analysis
Volume31
Issue number3
DOIs
StatePublished - 2000

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Composite materials
  • Elliptic regularity

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