The Neumann problem for higher order elliptic equations with symmetric coefficients

Ariel Barton; Steve Hofmann; Svitlana Mayboroda

doi:10.1007/s00208-017-1606-3

The Neumann problem for higher order elliptic equations with symmetric coefficients

Ariel Barton, Steve Hofmann, Svitlana Mayboroda

Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

In this paper we establish well posedness of the Neumann problem with boundary data in L² or the Sobolev space W˙-12, 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.

Original language	English (US)
Pages (from-to)	297-336
Number of pages	40
Journal	Mathematische Annalen
Volume	371
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-017-1606-3
State	Published - Jun 1 2018

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2017, Springer-Verlag GmbH Deutschland.

Keywords

31B10
35C15
35J30

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N2 - In this paper we establish well posedness of the Neumann problem with boundary data in L2 or the Sobolev space W˙-12, 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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The Neumann problem for higher order elliptic equations with symmetric coefficients

Abstract

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