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.