What does $\inf\int$ mean? 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.
 A: $$\inf \int f(x)dx$$ 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
A: 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 $u$.
A: 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.
