### Abstract

A family of linear homogeneous 2^{nd} order strongly elliptic symmetric systems with real constant coefficients, and bounded nonsmooth convex domains Ω are constructed in ℝ^{6} so that the systems have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces W^{1,2}(Ω). The construction is deduced from the model construction for a 4^{th} 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 language | English (US) |
---|---|

Pages (from-to) | 945-965 |

Number of pages | 21 |

Journal | Mathematical Research Letters |

Volume | 22 |

Issue number | 3 |

State | Published - 2015 |

### Fingerprint

### 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

**A constant coefficient Legendre-Hadamard system with no coercive constant coefficient quadratic form over W ^{1,2}.** / Verchota, Gregory.

Research output: Contribution to journal › Article

^{1,2}',

*Mathematical Research Letters*, vol. 22, no. 3, pp. 945-965.

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TY - JOUR

T1 - A constant coefficient Legendre-Hadamard system with no coercive constant coefficient quadratic form over W1,2

AU - Verchota, Gregory

PY - 2015

Y1 - 2015

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

UR - http://www.scopus.com/inward/record.url?scp=84929853408&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929853408&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84929853408

VL - 22

SP - 945

EP - 965

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 3

ER -