This is Exercise 8.22 from John Hunters Applied Analysis books. The question says:
Consider the non-convex functional:
Where W is the Sobolev space of functions that belong to $L^4([0,1])$ and whose weak derivatives belong to $L^4([0,1])$. Show that the infimum of f on W is equal to zero, but that the infimum is not attained
I have to show the following:
- Prove that f is not convex
- What is the infimum of f on W
- Is the infimum attained and explain
For part 1) I know the definition of what a convex function is, but this is a functional and I'm not sure on how to show this.
For part 2) I'm trying to use a method used in another similar problem stated here:
However, I can't seem to get it to work. I'm trying to use a triangle function, but the problem is that I can't make the integral equal to zero as easily as they have done in their problem. I can almost do it except for the $v^2$ term in my integrand.
Also if I pick my $v(x)$ to be a constant such as $v(x)=c$, it's true that I can find a value of c that makes the overall integral zero, yet c will be c=i which is an imaginary answer and I don't think this is allowed in my domain.
Any help would be great.