Fundamental Lemma of Calculus of Variations - The Du Bois-Reymond Lemma

The Du Bois-Reymond Lemma

The du Bois-Reymond lemma (named after Paul du Bois-Reymond) is a more general version of the above lemma. It defines a sufficient condition to guarantee that a function vanishes almost everywhere. Suppose that is a locally integrable function defined on an open set . If

for all then f(x) = 0 for almost all x in Ω. Here, is the space of all infinitely differentiable functions defined on Ω whose support is a compact set contained in Ω.