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

Gregory C. Verchota (Department of Mathematics, Syracuse University, Syracuse, New York, U.S.A.)

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

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