fundamental lemma of calculus of variations

The idea in the calculus of variations is to studystationary points of functionals.To derive a differential equation for such stationarypoints, the following theorem is needed, and hencenamed thereafter. It is also used in distribution theoryto recover traditional calculus from distributional calculus.

Theorem 1.

Suppose $f\mathrm{:}U\mathrm{\to }\mathrm{C}$ is a locally integrable function on anopen 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 toprove the claim for real $f$. If $f$ is continuous, this can be seenby purely geometrical arguments. A full proofbased on the Lebesgue differentiation theorem is givenin [1]. Another proof is given in [2].