Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

Ariel Barton; Steve Hofmann; Svitlana Mayboroda

doi:10.5802/aif.3278

Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

Ariel Barton, Steve Hofmann, Svitlana Mayboroda

Mathematics

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

5 Scopus citations

Abstract

We show that if u is a solution to a linear elliptic differential equation of order 2m > 2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space L^p(Rⁿ) or Sobolev space W _± ^p ₁(Rⁿ). Even in the case where u is a solution to a second order equation, our results are new for certain values of p.

Original language	English (US)
Pages (from-to)	1627-1678
Number of pages	52
Journal	Annales de l'Institut Fourier
Volume	69
Issue number	4
DOIs	https://doi.org/10.5802/aif.3278
State	Published - 2020

Bibliographical note

Funding Information:
Keywords: Elliptic equation, higher order differential equation, Dirichlet boundary values, Neumann boundary values. 2010 Mathematics Subject Classification: 35J67, 35J30, 31B10. (*) Steve Hofmann is partially supported by the NSF grant DMS-1664047. Svitlana Mayboroda is partially supported by the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, the NSF Materials Research Science and Engineering Center Seed Grant, and the Simons Fellowship.

Publisher Copyright:
© Association des Annales de l’institut Fourier, 2019, Certains droits réservés.

Keywords

Dirichlet boundary values
Elliptic equation
Higher order differential equation
Neumann boundary values

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title = "Dirichlet and Neumann boundary values of solutions to higher order elliptic equations",

abstract = "We show that if u is a solution to a linear elliptic differential equation of order 2m > 2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space Lp(Rn) or Sobolev space W ± p 1(Rn). Even in the case where u is a solution to a second order equation, our results are new for certain values of p.",

keywords = "Dirichlet boundary values, Elliptic equation, Higher order differential equation, Neumann boundary values",

author = "Ariel Barton and Steve Hofmann and Svitlana Mayboroda",

note = "Funding Information: Keywords: Elliptic equation, higher order differential equation, Dirichlet boundary values, Neumann boundary values. 2010 Mathematics Subject Classification: 35J67, 35J30, 31B10. (*) Steve Hofmann is partially supported by the NSF grant DMS-1664047. Svitlana Mayboroda is partially supported by the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, the NSF Materials Research Science and Engineering Center Seed Grant, and the Simons Fellowship. Publisher Copyright: {\textcopyright} Association des Annales de l{\textquoteright}institut Fourier, 2019, Certains droits r{\'e}serv{\'e}s.",

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AU - Barton, Ariel

AU - Hofmann, Steve

AU - Mayboroda, Svitlana

N1 - Funding Information: Keywords: Elliptic equation, higher order differential equation, Dirichlet boundary values, Neumann boundary values. 2010 Mathematics Subject Classification: 35J67, 35J30, 31B10. (*) Steve Hofmann is partially supported by the NSF grant DMS-1664047. Svitlana Mayboroda is partially supported by the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, the NSF Materials Research Science and Engineering Center Seed Grant, and the Simons Fellowship. Publisher Copyright: © Association des Annales de l’institut Fourier, 2019, Certains droits réservés.

PY - 2020

Y1 - 2020

N2 - We show that if u is a solution to a linear elliptic differential equation of order 2m > 2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space Lp(Rn) or Sobolev space W ± p 1(Rn). Even in the case where u is a solution to a second order equation, our results are new for certain values of p.

AB - We show that if u is a solution to a linear elliptic differential equation of order 2m > 2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space Lp(Rn) or Sobolev space W ± p 1(Rn). Even in the case where u is a solution to a second order equation, our results are new for certain values of p.

KW - Dirichlet boundary values

KW - Elliptic equation

KW - Higher order differential equation

KW - Neumann boundary values

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JF - Annales de l'Institut Fourier

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ER -

Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

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