Preprint 64/2002

The set of gradients of a bump

Jan Kolár and Jan Kristensen

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Submission date: 02. Aug. 2002
Pages: 24
Bibtex
MSC-Numbers: 26B05, 26B30, 46G05, 46B20
Keywords and phrases: gradient, range of derivative, bump, morse-sard theorem, critical set
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Abstract:
This is the first in a series of two papers.

The range of the gradient of a differentiable real-valued function with a non-empty and bounded support (i.e., a bump) is investigated. For a smooth bump formula20 on formula22 it is shown that the interior formula24 is connected and dense in formula26. A purely topological characterization of such gradient ranges is however impossible. We give an example of a compact set formula28 that is homeomorphic to the closed unit disk, but such that no affine image of K is the gradient range of a smooth bump on formula22. For smooth bumps on formula34 we show that the gradient range cannot be separated by a totally disconnected set. The proof relies on a Morse-Sard type result involving irreducible separators of formula34. Proofs are carried out for a class of formula38 functions containing all those whose first order derivatives are Lipschitz or of bounded variation.

Finally, we present an example of a formula38-smooth bump on formula42, which has a gradient range with non-empty and disconnected interior, and a formula44-smooth weak bump on formula42 with the same property.

03.04.2017, 12:08