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Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is $$M \equiv\int_{x=-\infty}^\infty u(x) ~ dx$$ where $M>0$.

I want to find the minimum value of $$E[u] = \int_{x=-\infty}^\infty u^3 -u^2 +(u_x)^2 ~ dx$$ and ideally the function $u$ that minimizes $E$.

My conjecture is that it is an elliptic function with compact support, but before I dig deep into the analysis I am wondering if the answer is known.

This question arises in determining the attractors of a degenerate Cahn-Hilliard Equation, $$u_t = - [u(u_{xx} +3u^2 -2u)_x]_x $$ which admits non-negative solutions $u(x,t)$ with compact support. The attractors at large time yields the problem above.

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Probably this is well known, but I'm personally not sure to whom. I think your conjecture is very likely to be correct and should follow straight forwardly from using energy conservation to solve the corresponding Euler-Lagrange equation (minimization condition). You'll have a first order ODE that is solvable by separation of variables, with $x$ ultimately given by an elliptic integral wrt to $u$. Inversion naturally gives $u(x)$ in terms of elliptic functions. –  Igor Khavkine Jun 6 '13 at 21:12
    
Igor, I believe you get a second order ODE of the form $-u_{xx} + 3u^2 - 2u + c = 0$. Also, I'm also not sure what you mean by separation of variables, since it's an ODE and not a PDE. –  Deane Yang Jun 6 '13 at 22:22
    
Are you sure you have $u^3$ and not $u^4$ in the functional? I'm saying because Cahn-Hilliard is the gradient flow of the functional with $u^4$. The minimizers of that functional (with $u^4$) are well understood (Google for "De Giorgi's conjecture") and are definitely not compactly supported. If you stick to $u^3$, the equation you have is the one that you would get for traveling waves in the KPP-Fisher equation. Those are well understood as well. They are certainly not compactly supported either. –  Luis Silvestre Jun 6 '13 at 22:55
    
@Deane, the equation is time translation invariant. So one can directly write down the first integral, $E=(u_x)^2-cu+u^2-u^3$, where $E$ is some integration constant. This is a first order ODE, which can be converted to the form $dx = du/f(u)$. That is what I meant. In this particular problem, $f(u) = \sqrt{E+cu-u^2+u^3}$, which leads to elliptic integrals. –  Igor Khavkine Jun 7 '13 at 10:18
    
@Igor -Yes, this is where the elliptic functions come from. The obstruction is the mass doesn't come out as a nice function of E or c. –  shrdlu Jun 7 '13 at 13:52

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