Type-changing PDE and singularities of Monge characteristic systems. (English) Zbl 1479.35578

Noda, Takahiro; Shibuya, Kazuhiro

Shoda, Toshihiro (ed.) et al., Differential geometry and Tanaka theory. Differential system and hypersurface theory. Proceedings of the international conference, RIMS, Kyoto University, Japan, January, 24–28, 2011. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 82, 57-73 (2019).

The authors study second order type-changing partial differential equations on \(\mathbb R^2\), with particular focus on the parabolic points. At each point in its domain, a second order partial differential equation is either elliptic, parabolic or hyperbolic and ‘type-changing’ refers to the fact that the equation attains at least two of these types. In a previous paper [Osaka J. Math. 49, No. 1, 101–124 (2012; Zbl 1246.35137)], the authors identified three types of parabolic points, characterized by the fact that an associated differential system is either ‘involute’, of ‘finite type’ or of ‘torsion type’. In the present paper, the authors introduce the notion of Monge characteristic system as a subset of the 2-jet space over the set of parabolic points and prove that a parabolic point \(p\) is involute if the rank of the Monge characteristic system above \(p\) is 2 and either of finite type or of torsion type if the rank of the Monge characteristic system above \(p\) is 1.
For the entire collection see [Zbl 1437.53042].

Reviewer: Jakob Hultgren (Washington)

MSC:

35N10	Overdetermined systems of PDEs with variable coefficients
35M10	PDEs of mixed type

Keywords:

type-changing equations; regular overdetermined systems; structure equations; Monge characteristic systems; parabolic points

Citations:

Zbl 1246.35137

References:

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