## Foliations and separation of variables.(English)Zbl 0553.58037

Structure transverse des feuilletages, Toulouse 1982, Astérisque 116, 214-221 (1984).
[For the entire collection see Zbl 0534.00014.]
Let L be a linear partial differential operator, U and V be differential ideals defining a pair of complementary foliations, and $$F_ U$$ and $$F_ V$$ be the spaces of functions constant on the leaves of the two foliations. Then U and V are said to split L if there is a positive function R and partial differential operators $$M: F_ U\to F_ U$$ and $$N: F_ V\to F_ V$$ such that for all $$u\in F_ U$$ and $$v\in F_ V$$ $$L(uv)=R(vM(u)+uN(v)).$$ The author gives necessary and sufficient conditions for a given pair of foliations to split L, and shows by examples how to apply the conditions to find all splittings of a given operator. He also discusses the simultaneous splitting of a 1-parameter family of operators.
Reviewer: B.Reinhart

### MSC:

 58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 57R30 Foliations in differential topology; geometric theory

Zbl 0534.00014