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Variational derivatives and p ‐gradients of functionals on spaces of continuously differentiable functions
Author(s) -
Hamilton E. P.,
Nashed M. Z.,
Brosowski B.
Publication year - 1983
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670050132
Subject(s) - smoothness , mathematics , differentiable function , calculus of variations , context (archaeology) , derivative (finance) , boundary (topology) , variational analysis , directional derivative , mathematical analysis , boundary value problem , euler–lagrange equation , variational principle , lagrangian , paleontology , financial economics , economics , biology
The purpose of this paper is to introduce and study a new type of derivative – the variational gradient – for a functional on C n [ a, b ]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on C n [ a, b ] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler‐Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations. Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.