Fundamental lemma of the calculus of variations

Remarks

For continuous $f$ the proof is short. If $f(x_0) > 0$, then $f > 0$ on a ball $B$ around $x_0$; a bump function $\varphi \ge 0$ supported in $B$ with $\varphi(x_0) > 0$ gives $\int_\Omega f \varphi \, dx > 0$, a contradiction. For locally integrable $f$, one convolves $f$ with a smooth mollifier: the hypothesis makes every mollified function vanish, and these converge to $f$ in $L^1$ on compact sets.

The lemma is the step that turns the vanishing of the first variation into the Euler–Lagrange equation. After an integration by parts, the first variation has the form $\int_\Omega E(x) \varphi(x) \, dx$, where $E$ is the Euler–Lagrange expression; the lemma gives $E = 0$.

The class of test functions must be large enough. If only $\varphi$ with $\int_\Omega \varphi \, dx = 0$ were allowed, every constant $f$ would satisfy the hypothesis; this case is the du Bois-Reymond lemma.

Going deeper

For a student

In a derivation of the Euler–Lagrange equation one first finds $\int_a^b \big(L_z - \frac{d}{dx} L_p\big)\eta \, dx = 0$ for all $\eta$ that vanish at the ends. The lemma then gives the pointwise equation $L_z - \frac{d}{dx} L_p = 0$. In distribution theory the lemma says that a locally integrable function is determined by the distribution it defines.