Weak minimum vs. strong minimum

Question

Consider the problem of minimizing the functional $$ J(y) = \int_0^1 y'(x)^2(1-y'(x)^2)dx $$ with the end-points condition $y(0) = y(1) = 0$. Is the curve $y \equiv 0$ a weak minimum or a strong minimum of $J$?

Clearly, $y$ is a weak minimum. How can I make a conclusion about $y$ being a strong minimum or not?

Answer 1

Look at $f'_{\epsilon,\delta}$ given by the periodic extension of the piecewise linear interpolant of $(0,2),(\epsilon,2),(\epsilon+\delta,-2),(2\epsilon+\delta,-2),(2\epsilon+2\delta,2)$, where $\epsilon$ and $\delta$ are chosen so that $1/(2\epsilon+2\delta)$ is an integer. Then $f_{\epsilon,\delta}(x):=\int_0^x f'_{\epsilon,\delta}(y) dy$ will:

• Be $C^1$.
• Satisfy the boundary conditions (this is why we needed the periodicity structure).
• Have $C^0$ norm on the order of $\epsilon$ provided $\delta$ is comparable to $\epsilon$ or smaller.
• Have $J=-12+O(\delta)<0$ for small $\delta$ at fixed $\epsilon$. Here $-12$ is what you would get if the derivative simply bounced back and forth between $2$ and $-2$ without changing continuously in between.

This example (when the details are ironed out) will prove that $y=0$ is not a strong minimum of $J$. The idea of it is that a function with a small $C^0$ norm, even if it is $C^1$, is free to wiggle a great deal, as long as its wiggles always bring it back towards zero.