We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X,Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F'(X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear: we discuss the existence of a mapping F from a Banach space X into a Banach space Y, which is bounded, Lipschitz-continuous, and so that for all x, y ∈ X, if x ≠ y, then ||F'(x) - F'(y)||L(X,Y) > 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations.
@article{urn:eudml:doc:41643,
title = {On the range of the derivative of a smooth function and applications.},
journal = {RACSAM},
volume = {100},
year = {2006},
pages = {63-74},
mrnumber = {MR2267401},
zbl = {1109.46042},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41643}
}
Deville, Robert. On the range of the derivative of a smooth function and applications.. RACSAM, Tome 100 (2006) pp. 63-74. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41643/