A constant coefficient Legendre-Hadamard system with no coercive constant coefficient quadratic form over W1,2

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Abstract

A family of linear homogeneous 2nd order strongly elliptic symmetric systems with real constant coefficients, and bounded nonsmooth convex domains Ω are constructed in ℝ6 so that the systems have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces W1,2(Ω). The construction is deduced from the model construction for a 4th order scalar case [Ver14]. The latter is stated and parts of its proof discussed, one particular being the utility of having noncoercive formally positive forms as a starting point. An application of Macaulay's determinantal ideals to the noncoerciveness of formally positive forms for systems is then given.

Original languageEnglish (US)
Pages (from-to)945-965
Number of pages21
JournalMathematical Research Letters
Volume22
Issue number3
StatePublished - 2015

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Legendre
Quadratic form
Coefficient
Nonsmooth Domains
Convex Domain
Sobolev Spaces
Scalar
Form
Model

Keywords

  • Determinantal ideal
  • Indefinite form
  • Korn's inequality
  • Neumann problem
  • Null form
  • Rellich identity
  • Strongly elliptic
  • Sum of squares

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "A constant coefficient Legendre-Hadamard system with no coercive constant coefficient quadratic form over W1,2",
abstract = "A family of linear homogeneous 2nd order strongly elliptic symmetric systems with real constant coefficients, and bounded nonsmooth convex domains Ω are constructed in ℝ6 so that the systems have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces W1,2(Ω). The construction is deduced from the model construction for a 4th order scalar case [Ver14]. The latter is stated and parts of its proof discussed, one particular being the utility of having noncoercive formally positive forms as a starting point. An application of Macaulay's determinantal ideals to the noncoerciveness of formally positive forms for systems is then given.",
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T1 - A constant coefficient Legendre-Hadamard system with no coercive constant coefficient quadratic form over W1,2

AU - Verchota, Gregory

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N2 - A family of linear homogeneous 2nd order strongly elliptic symmetric systems with real constant coefficients, and bounded nonsmooth convex domains Ω are constructed in ℝ6 so that the systems have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces W1,2(Ω). The construction is deduced from the model construction for a 4th order scalar case [Ver14]. The latter is stated and parts of its proof discussed, one particular being the utility of having noncoercive formally positive forms as a starting point. An application of Macaulay's determinantal ideals to the noncoerciveness of formally positive forms for systems is then given.

AB - A family of linear homogeneous 2nd order strongly elliptic symmetric systems with real constant coefficients, and bounded nonsmooth convex domains Ω are constructed in ℝ6 so that the systems have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces W1,2(Ω). The construction is deduced from the model construction for a 4th order scalar case [Ver14]. The latter is stated and parts of its proof discussed, one particular being the utility of having noncoercive formally positive forms as a starting point. An application of Macaulay's determinantal ideals to the noncoerciveness of formally positive forms for systems is then given.

KW - Determinantal ideal

KW - Indefinite form

KW - Korn's inequality

KW - Neumann problem

KW - Null form

KW - Rellich identity

KW - Strongly elliptic

KW - Sum of squares

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