Fundamental lemma of the 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. If a continuous function on an open interval satisfies the equality for all compactly supported smooth functions on , then is identically zero. 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 (or and a number of its derivatives) vanishes at the endpoints , ; in this case the closed interval is used. 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. 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 smooth functions h on (a,b) such that , then g is constant. 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.