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 $f\mathrm{:}U\mathrm{\to }\mathrm{C}$ is a locally integrable function on an open subset $U\mathrm{\subset }{\mathrm{R}}^n$, and suppose that

$${\int }_{U}f\varphi 𝑑x=0$$

for all smooth functions with compact support $\varphi \mathrm{\in }{C}_{\mathrm{0}}^{\mathrm{\infty }}\mathit{}\mathrm{(}U\mathrm{)}$. Then $f\mathrm{=}\mathrm{0}$ almost everywhere.

By linearity of the integral, it is easy to see that one only needs to prove the claim for real $f$. If $f$ 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].

Title	fundamental lemma of calculus of variations
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