Some Calculus of Variations

Problem 1. Let $f : (a, b) \to \mathbb R$ be continuous. Prove that if $f \neq 0$ , then there exists $c, d \in (a, b)$ such that $\pm f(x) > 0$ whenever $x \in (c, d)$ .

(Click for Solution)

Solution. Suppose $f(x_0) \neq 0$ for some $x_0 \in (a, b)$ . Then $f(x_0) > 0$ or $f(x_0) < 0$ . In the case $f(x_0) > 0$ , use the continuity of $f$ to find $\delta \in (0, \min\{x_0-a, b-x_0\})$ such that

$\displaystyle |x - x_0| < \delta \quad \Rightarrow \quad |f(x) - f(x_0)| < \frac{f(x_0)}{2}.$

By expanding the inequality,

$\displaystyle f(x) - f(x_0) > -\frac{f(x_0)}{2} \quad \Rightarrow \quad f(x) > \frac{f(x_0)}{2} > 0.$

Defining $c := x_0 - \delta$ and $d:= x_0 + \delta$ ,

$x \in (c, d) \quad \Rightarrow \quad |x - x_0| < \delta \quad \Rightarrow \quad f(x) > 0.$

In the case $f(x_0) < 0$ , we have $(-f)(x_0) > 0$ . Applying the first case, there exists $c, d \in (a, b)$ such that

$x \in (c, d) \quad \Rightarrow \quad -f(x) = (-f)(x) > 0.$

Problem 2. Let $f : (a, b) \to \mathbb R$ be continuous and suppose $f(x) \geq 0$ for any $(a, b)$ . Prove that

$\displaystyle \int_a^b f(x)\, \mathrm dx = 0 \quad \Rightarrow \quad f = 0.$

(Click for Solution)

Solution. Suppose instead that $f \neq 0$ . Then there exists $(c, d) \subseteq (a, b)$ such that for any $x \in (c, d)$ , $f(x) > 0$ . Hence,

$\begin{aligned} \int_a^b f(x)\, \mathrm dx &= \int_a^c f(x)\, \mathrm dx + \int_c^d f(x)\, \mathrm dx + \int_d^b f(x)\, \mathrm dx \\ &\geq 0 + \int_c^d f(x)\, \mathrm dx + 0\\ &= \int_c^d f(x)\, \mathrm dx > 0 . \end{aligned}$

Definition 1. Call a function $h \in C^\infty$ compactly supported if there exists real numbers $c < d$ such that $h(x) = 0$ for $x \leq c$ and $x \geq d$ .

Problem 3. Let $f : (a, b) \to \mathbb R$ be continuous. Suppose that for any compactly supported $h$ ,

$\displaystyle \int_a^b f(x)h(x)\, \mathrm dx = 0.$

Then $f = 0$ .

(Click for Solution)

Solution. By way of contradiction, suppose $f \neq 0$ . By Problem 1, there exists $(c, d) \subseteq (a, b)$ such that without loss of generality, $f(x) > 0$ whenever $x \in (c, d)$ . Define $h \in C^\infty$ by

$h(x) = \begin{cases} e^{ -\frac{1}{ (x-c)(d-x) } }, & x \in (c, d), \\ 0, & \text{otherwise}.\end{cases}$

Then it is clear that $h$ is compactly supported, and that for any $x \in (a, b)$ , $f(x)h(x) \geq 0$ . By Problem 2, $f \cdot h = 0$ . However for $m := (c+d)/2$ ,