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

 $\int_{U}f\phi dx=0$

for all smooth functions with compact support $\phi\in C_{0}^{\infty}(U)$. Then $f=0$ 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].

## References

• 1 L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
• 2 S. Lang, Analysis II, Addison-Wesley Publishing Company Inc., 1969.
Title fundamental lemma of calculus of variations FundamentalLemmaOfCalculusOfVariations 2013-03-22 15:02:04 2013-03-22 15:02:04 matte (1858) matte (1858) 8 matte (1858) Theorem msc 28B15 fundamental theorem of the calculus of variations CalculusOfVariations