A relationship between the dirichlet and regularity problems for elliptic equations

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Abstract

Let ℒ = divA∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation ℒu = 0 in a bounded Lipschitz domain Ω of ℝn. We study the relationship between the solvability of the Lp Dirichlet problem (D)p with boundary data in Lp(∂Ω) and that of the Lq regularity problem (R)q with boundary data in W1,q(∂Ω), where 1 < p, q < ∞. It is known that the solvability of (R)p implies that of (D)p′. In this note we show that if (D)p′ is solvable, then either (R)p is solvable or (R)q is not solvable for any 1 < q < ∞.

Original languageEnglish
Pages (from-to)205-213
Number of pages9
JournalMathematical Research Letters
Volume14
Issue number2-3
DOIs
StatePublished - 2007

Keywords

  • Dirichlet problem
  • Elliptic equation
  • Regularity problem

ASJC Scopus subject areas

  • General Mathematics

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