fundamental lemma of calculus of variations

The idea in the calculus of variationsMathworldPlanetmath is to study stationary points of functionals. To derive a differential equationMathworldPlanetmath 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:UC is a locally integrable function on an open subset URn, and suppose that


for all smooth functions with compact support ϕC0(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 continuousMathworldPlanetmath, 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.
Title fundamental lemma of calculus of variationsMathworldPlanetmathPlanetmath
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