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].

Bibliography

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.