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# Mathematical Research Letters

## Volume 22 (2015)

### Number 3

### A constant coefficient Legendre-Hadamard system with no coercive constant coefficient quadratic form over $W^{1,2}$

Pages: 945 – 965

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n3.a16

#### Author

#### Abstract

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

#### Keywords

Neumann problem, strongly elliptic, Korn’s inequality, sum of squares, null form, indefinite form, Rellich identity, determinantal ideal