1
$\begingroup$

Maybe this is a strange or un-professional question. I want to know whether my equation is convex.

My equation is as follows: $E\left(\phi\right)=\int_{\Omega}\left|\left(g\left(\overline{x}\right)-A\right)\right|^{2}\left(1-H(\phi\left(\overline{x}\right))\right)d\overline{x} +\mu\int_{\Omega}H(\phi\left(\overline{x})\right)d\overline{x} +\nu\int_{\Omega}\delta\left(\phi\left(\overline{x}\right)\right)\left|\nabla\phi\left(\overline{x}\right)\right|d\overline{x}$

where $g$ is a known function $R^{2}\rightarrow R$, and A is a know scalar, $H$ is a Heaviside function,e.g. $H\left(\phi\right)=\frac{1}{2}+\frac{1}{\pi}\arctan\left(\phi\right)$ and $\delta$ is a Delta function, I want to find the $\phi,R^{2}\rightarrow R$ which minimize $E$. So I want to know whether $E$ is convex. So if I find its extremum, then I find its extrernes.

  • 0
    I am not sure if that will be called Convex functional Analysis.2012-02-05
  • 2
    At this level of generality, all I can suggest is that you use the definition of "convex". In most cases I've encountered, it would be fairly easy to decide. We might be able to help if you gave us some specific details about the particular problem.2012-02-05
  • 0
    Right now, this question is on its 14th edit, and looks completely different from its original form. Please open a new question if you have something new to ask, instead of changing the question entirely, as it makes the existing comments and answers seem irrelevant. See [this meta thread](http://meta.math.stackexchange.com/q/2814/856) for more info.2012-02-10
  • 0
    Thank you so much for your advise. I will turn the original question back.2012-02-10

2 Answers 2