Fundamental lemma of calculus of variations

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

Basic versionEdit

If a continuous function   on an open interval   satisfies the equality
 
for all compactly supported smooth functions   on  , then   is identically zero.[1][2]

Here "smooth" may be interpreted as "infinitely differentiable",[1] but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous",[2] since these weaker statements may be strong enough for a given task. "Compactly supported" means "vanishes outside   for some  ,   such that  ";[1] but often a weaker statement suffices, assuming only that   (or   and a number of its derivatives) vanishes at the endpoints  ,  ;[2] in this case the closed interval   is used.

Version for two given functionsEdit

If a pair of continuous functions f, g on an interval (a,b) satisfies the equality
 
for all compactly supported smooth functions h on (a,b), then g is differentiable, and g' = f  everywhere.[3][4]

The special case for g = 0 is just the basic version.

Here is the special case for f = 0 (often sufficient).

If a continuous function g on an interval (a,b) satisfies the equality
 
for all compactly supported smooth functions h on (a,b), then g is constant.[5]

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 functionsEdit

The given functions (f, g) may be discontinuous, provided that they are locally integrable (on the given interval). In this case, Lebesgue integration is meant, the conclusions hold almost everywhere (thus, in all continuity points), and differentiability of g is interpreted as local absolute continuity (rather than continuous differentiability).[6][7] 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.[4]

Higher derivativesEdit

If a tuple of continuous functions   on an interval (a,b) satisfies the equality
 
for all compactly supported smooth functions h on (a,b), then there exist continuously differentiable functions   on (a,b) such that
 
everywhere.[8]

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 functionsEdit

Generalization to vector-valued functions   is straightforward; one applies the results for scalar functions to each coordinate separately,[9] or treats the vector-valued case from the beginning.[10]

Multivariable functionsEdit

If a continuous multivariable function f on an open set   satisfies the equality
 
for all compactly supported smooth functions h on Ω, then f is identically zero.

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 (rather than compactly supported).[11]

Here is a version for discontinuous multivariable functions.

Let   be an open set, and   satisfy the equality
 
for all compactly supported smooth functions h on Ω. Then f=0 (in L2, that is, almost everywhere).[12]

ApplicationsEdit

This lemma is used to prove that extrema of the functional

 

are weak solutions   (for an appropriate vector space  ) of the Euler–Lagrange equation

 

The Euler–Lagrange equation plays a prominent role in classical mechanics and differential geometry.

NotesEdit

  1. ^ a b c Jost & Li-Jost 1998, Lemma 1.1.1 on p.6
  2. ^ a b c Gelfand & Fomin 1963, Lemma 1 on p.9 (and Remark)
  3. ^ Gelfand & Fomin 1963, Lemma 4 on p.11
  4. ^ a b Hestenes 1966, Lemma 15.1 on p.50
  5. ^ Gelfand & Fomin 1963, Lemma 2 on p.10
  6. ^ Jost & Li-Jost 1998, Lemma 1.2.1 on p.13
  7. ^ Giaquinta & Hildebrandt 1996, section 2.3: Mollifiers
  8. ^ Hestenes 1966, Lemma 13.1 on p.105
  9. ^ Gelfand & Fomin 1963, p.35
  10. ^ Jost & Li-Jost 1998
  11. ^ Gelfand & Fomin 1963, Lemma on p.22; the proof applies in both situations.
  12. ^ Jost & Li-Jost 1998, Lemma 3.2.3 on p.170

ReferencesEdit

  • Jost, Jürgen; Li-Jost, Xianqing (1998), Calculus of variations, Cambridge University
  • Gelfand, I.M.; Fomin, S.V. (1963), Calculus of variations, Prentice-Hall (transl. from Russian).
  • Hestenes, Magnus R. (1966), Calculus of variations and optimal control theory, John Wiley
  • Giaquinta, Mariano; Hildebrandt, Stefan (1996), Calculus of Variations I, Springer