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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.

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