Whether something is defined or not is a matter of, well, definition. Division by zero is undefined because we explicitly exclude it from the definition of division. The reasons we exclude it from the definition are varied, of course, but it's not a matter of lack of knowledge.
It is not quite clear what you mean by "provide a value"; there are numbers which we can prove cannot be explicitly described in terms of a terminating algorithm (that is, there is no Turing Machine that will produce the number). But does that mean we do not provide a value?
We cannot write down exactly a number that solves the equation $x^2-2=0$. We cheat when we say the solutions are $\sqrt{2}$ and $-\sqrt{2}$ because... what does "$\sqrt{2}$" mean? It means "the positive real number that is a solution to $x^2-2=0$". Does that mean we "don't know how to provide a value"?
On the other hand, there are equations which we may genuinely not know whether they have solutions of a special kind or not. For a long time, it was unknown whether there were any positive integers $a$, $b$, and $c$, and a positive integer $n\gt 2$, such that $a^n+b^n=c^n$. Now we know there are none.