fundamental lemma of calculus of variations

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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.

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].

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.

Synonym:

fundamental theorem of the calculus of variations

Major Section:

Reference

Type of Math Object:

Theorem

Parent: