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On the Equation ∇u = uV

✍ Scribed by Holger Teismann

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
164 KB
Volume
254
Category
Article
ISSN
0022-247X

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✦ Synopsis

The PDE ∇u = uV can be treated as its ODE equivalent u = a u if the vector field V satisfies the integrability condition curl V = 0. In this case there exist solutions v of the equation ∇v = V and solutions of the original equation have the form u = ce v (c ∈ ). However, in devising meaningful solution concepts, the equation displays some surprising features related to local and global integrability of its solutions. In the course of the investigation technical tools like a lifting theorem with respect to the exponential map and Poincaré type theorems are proved for distributions and functions of Beppo Levi type.

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are presented. The proofs are based on the alternative method, a connectedness result, the contraction mapping principle, and a detailed analysis of the bifurcation equation utilizing, e.g., a generalization of the mean value theorem for integrals. We shall obtain results with g bounded or unbounded