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Overdetermined elliptic systems. (English) Zbl 1101.35061
The article discusses the notion of ellipticity for overdetermined systems. Naive application of the standard definition leads already to problems, if one considers a first-order form of Laplace’s equations. Traditionally, this is resolved by using an approach based on a weighted symbol leading to the concept of DN-ellipticity (after Douglis and Nirenberg).
Here it is shown that a much simpler solution lies in the completion to involution, as the origin of the problem are hidden integrability conditions. More precisely, it is proven that any system that is DN-elliptic becomes elliptic in the standard sense during its completion to involution. The converse is not true, as an explicit counter example demonstrates.

35N10 Overdetermined systems of PDEs with variable coefficients
35J45 Systems of elliptic equations, general (MSC2000)
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