On the range of the derivative of a smooth function and applications.

Deville, Robert. "On the range of the derivative of a smooth function and applications.." RACSAM 100.1-2 (2006): 63-74. <http://eudml.org/doc/41643>.

@article{Deville2006,
abstract = {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) &gt; 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations.},
author = {Deville, Robert},
journal = {RACSAM},
language = {eng},
number = {1-2},
pages = {63-74},
title = {On the range of the derivative of a smooth function and applications.},
url = {http://eudml.org/doc/41643},
volume = {100},
year = {2006},
}

TY - JOUR
AU - Deville, Robert
TI - On the range of the derivative of a smooth function and applications.
JO - RACSAM
PY - 2006
VL - 100
IS - 1-2
SP - 63
EP - 74
AB - 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) &gt; 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations.
LA - eng
UR - http://eudml.org/doc/41643
ER -