fundamental lemma of calculus of variations
The idea in the calculus of variations is to study
stationary points of functionals.
To derive a differential equation
for such stationary
points, the following theorem is needed, and hence
named thereafter. It is also used in distribution theory
to recover traditional calculus from distributional calculus.
Theorem 1.
Suppose is a locally integrable function on an open subset , and suppose that
for all smooth functions with compact support . Then almost everywhere.
By linearity of the integral, it is easy to see that one only needs to
prove the claim for real . If is continuous, this can be seen
by purely geometrical arguments. A full proof
based on the Lebesgue differentiation theorem is given
in [1]. Another proof is given in [2].
References
Title | fundamental lemma of calculus of variations![]() |
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Canonical name | FundamentalLemmaOfCalculusOfVariations |
Date of creation | 2013-03-22 15:02:04 |
Last modified on | 2013-03-22 15:02:04 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 8 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 28B15 |
Synonym | fundamental theorem of the calculus of variations |
Related topic | CalculusOfVariations |