Fundamental lemma of calculus of variations


In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point.
Accordingly, the necessary condition of extremum appears in a weak formulation integrated with an arbitrary function. The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation, free of the integration with arbitrary function. The proof usually exploits the possibility to choose concentrated on an interval on which keeps sign. Several versions of the lemma are in use. Basic versions are easy to formulate and prove. More powerful versions are used when needed.

Basic version

Here "smooth" may be interpreted as "infinitely differentiable", but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous", since these weaker statements may be strong enough for a given task. "Compactly supported" means "vanishes outside for some, such that "; but often a weaker statement suffices, assuming only that vanishes at the endpoints, ; in this case the closed interval is used.

Version for two given functions

The special case for g = 0 is just the basic version.
Here is the special case for f = 0.
If, in addition, continuous differentiability of g is assumed, then integration by parts reduces both statements to the basic version; this case is attributed to Joseph-Louis Lagrange, while the proof of differentiability of g is due to Paul du Bois-Reymond.

Versions for discontinuous functions

The given functions may be discontinuous, provided that they are locally integrable. In this case, Lebesgue integration is meant, the conclusions hold almost everywhere, and differentiability of g is interpreted as local absolute continuity. Sometimes the given functions are assumed to be piecewise continuous, in which case Riemann integration suffices, and the conclusions are stated everywhere except the finite set of discontinuity points.

Higher derivatives

This necessary condition is also sufficient, since the integrand becomes
The case n = 1 is just the version for two given functions, since and thus,
In contrast, the case n=2 does not lead to the relation since the function need not be differentiable twice. The sufficient condition is not necessary. Rather, the necessary and sufficient condition may be written as for n=2, for n=3, and so on; in general, the brackets cannot be opened because of non-differentiability.

Vector-valued functions

Generalization to vector-valued functions is straightforward; one applies the results for scalar functions to each coordinate separately, or treats the vector-valued case from the beginning.

Multivariable functions

Similarly to the basic version, one may consider a continuous function f on the closure of Ω, assuming that h vanishes on the boundary of Ω.
Here is a version for discontinuous multivariable functions.

Applications

This lemma is used to prove that extrema of the functional
are weak solutions of the Euler–Lagrange equation
The Euler–Lagrange equation plays a prominent role in classical mechanics and differential geometry.