How to check if a function is minimum to functional?

Question

Given $\int_0^1(y')^3dx$ functional and $y(0) = 0 ,y(1)=1$ conditions. Using Euler–Lagrange equation I have got $y(x)=x$. So $y$ is a stationary point of the functional. How to check if it is the minimum for $y \in C^2[0,1]$ ?

Answer 1

Consider the function $$ y(t) = \begin{cases} -\lambda t & \text{ for $t <1/\lambda$} \\\\ \frac{2}{\lambda-1} (\lambda t-1) -1 & \text{for $t\ge 1/\lambda$}. \end{cases} $$ You have $$ \int_0^1 (y')^3\, dx = -\lambda^2 + (1-1/\lambda)\left(\frac{2\lambda}{\lambda-1}\right)^3 \to -\infty \qquad \lambda\to +\infty $$ hence your functional does not have an absolute minimum.

It is possible to smooth out the function $y(t)$ to get a $C^\infty$ function with the same property.