What does $\inf\int$ mean?

Question

What does notation

$$\inf\int f(x) \,\mathrm{d}x$$

stand for? I noticed it in a question on this site.

Any keywords, links or stories about this or similar notations will be appreciated :)

Sorry for such a basic question, but i can't find a reasonable keyword to look this up anywhere.

Answer 1

It means infimum, which is a generalization of the notion of minimum. For example, $f(x)=1/x$ doesn't have a minimum on $x\in[1,\infty)$, but its infimum is $0$.

In the context in which it was used,

$$\inf \iint\limits_{x^2+y^2\leqslant1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\mathrm dx\,\mathrm dy$$ for $C^\infty$ functions $u$ that... [more conditions on $u$]

it means the tightest bound on how small the value of the integral can get for any such function $u$.

Answer 2

$$\inf \int f(x)dx$$ is the largest real number that is less than or equal to $$\int f(x) dx$$ inf stands for infimum. See more information here:

http://en.wikipedia.org/wiki/Infimum

Answer 3

To write $\displaystyle\inf\int f(x)\,dx$ without the words that came after that fails to convey what was said. It says "for $C^\infty$ functions that vanish at $0$ and [. . . . .]", and the expression inside the integral has "$u$" in it. In other words, it identifies a particular set. An infimum is an infimum of a set.

The infimum of a set with a smallest member is the smallest member. Thus the infimum of the set of all nonnegative numbers is $0$. The infimum of some sets with no smallest member exists. For example, there is no smallest positive number, and the infimum of the set of all positive numbers is $0$.

The infimum of a set is the greatest lower bound of the set. $0$ is a lower bound of the set of all positive numbers because $0$ is less than or equal to every positive number. No number bigger than $0$ is a lower bound of the set of all positive numbers. So $0$ is the greatest among all lower bounds.