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Mathematische Annalen

, Volume 371, Issue 1–2, pp 297–336 | Cite as

The Neumann problem for higher order elliptic equations with symmetric coefficients

  • Ariel BartonEmail author
  • Steve Hofmann
  • Svitlana Mayboroda
Article

Abstract

In this paper we establish well posedness of the Neumann problem with boundary data in \(L^2\) or the Sobolev space \(\dot{W}^2_{-1}\), in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for an elliptic divergence form higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space.

Mathematics Subject Classification

35J30 31B10 35C15 

Notes

Acknowledgements

We would like to thank the American Institute of Mathematics for hosting the SQuaRE workshop on “Singular integral operators and solvability of boundary problems for elliptic equations with rough coefficients,” and the Mathematical Sciences Research Institute for hosting a Program on Harmonic Analysis, at which many of the results and techniques of this paper were discussed.

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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, 309 SCENUniversity of ArkansasFayettevilleUSA
  2. 2.University of MissouriColumbiaUSA
  3. 3.Department of MathematicsUniversity of MinnesotaMinneapolisUSA

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