fundamental lemma of calculus of variationsTheoremForLocallyIntegrableFunctions2005-02-14 16:03:012005-11-04 15:18:10Theoremlocally integrable function% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thm}{Theorem}
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\newtheorem{cor}{Corollary}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.
\begin{thm}
Suppose $f\colon U\to \C$ is a locally integrable function on an
open subset $U\subset \sR^n$,
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.
\end{thm}
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 \cite{hormander}. Another proof is given in \cite{lang}.
\begin{thebibliography}{9}
\bibitem{hormander}
L. H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I,
(Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990.
\bibitem{lang}
S. Lang, \emph{Analysis II},
Addison-Wesley Publishing Company Inc., 1969.
\end{thebibliography}