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The Neumann problem for higher order elliptic equations with symmetric coefficients

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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.

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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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Authors and Affiliations

  1. Department of Mathematical Sciences, 309 SCEN, University of Arkansas, Fayetteville, AR, 72701, USA

    Ariel Barton

  2. University of Missouri, 202 Math Sciences Bldg., Columbia, MO, 65211, USA

    Steve Hofmann

  3. Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Svitlana Mayboroda

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  1. Ariel Barton
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  2. Steve Hofmann
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  3. Svitlana Mayboroda
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Correspondence to Ariel Barton.

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Communicated by Loukas Grafakos.

Steve Hofmann is partially supported by the NSF Grant DMS-1361701. Svitlana Mayboroda is partially supported by the Simons Foundation, the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, and the NSF Materials Research Science and Engineering Center Seed Grant.

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Barton, A., Hofmann, S. & Mayboroda, S. The Neumann problem for higher order elliptic equations with symmetric coefficients. Math. Ann. 371, 297–336 (2018). https://doi.org/10.1007/s00208-017-1606-3

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