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Mathematische Annalen

, Volume 361, Issue 3–4, pp 863–907 | Cite as

The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

  • Steve Hofmann
  • Carlos Kenig
  • Svitlana MayborodaEmail author
  • Jill Pipher
Article

Abstract

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with \(t\)-independent complex bounded measurable coefficients (\(t\) being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in \(L^{p'}\), subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in \(L^p\). Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with \(t\)-independent real (possibly non-symmetric) coefficients there exists a \(p>1\) such that the Regularity problem is well-posed in \(L^p\).

Notes

Acknowledgments

Hofmann was partially supported by the NSF grant DMS 1101244. Kenig was partially supported by the NSF grants DMS 0968472 and DMS 1265249. Mayboroda was partially supported by the NSF grants DMS 1344235, DMS 1220089, DMR 0212302, and the Alfred P. Sloan Fellowship. Pipher was partially supported by the Australian Research Council grant ARC-DP120100399. This work has been possible thanks to the support and hospitality of the University of Chicago, the University of Minnesota, the University of Missouri, Brown University, the Institute for Computational and Experimental Research in Mathematics, and the American Institute of Mathematics. The authors would like to express their gratitude to these institutions. Finally, the authors would like to thank the referee for the careful reading of the manuscript and many helpful suggestions improving the exposition of the paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Steve Hofmann
    • 1
  • Carlos Kenig
    • 2
  • Svitlana Mayboroda
    • 3
    Email author
  • Jill Pipher
    • 4
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  4. 4.Department of MathematicsBrown UniversityProvidenceUSA

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