@PaulSiegel, I would be interested to know if you were inspired by some reference (book/paper).

Could you give some more detail on "Viscosity solutions are to weak solutions as the maximum principle is to energy methods" please? And also “So it makes sense to define your weak solution as a one for which the maximum principle holds when you compare to smooth functions”?

@HaraldHanche-Olsen, Oh right. So we kind of integrate $h(x) = \begin{cases} \infty & \text{if } x = 0,\\ f(0) & \text{if } x\in (-1,1),\\ \infty & \text{if } x =1. \end{cases}$ which gives us $\int_0^1h(x)dx=f(0)$, now I get it, thanks!

@HaraldHanche-Olsen, okay I see. I am just having some difficulty to apply this particular visualisation of Stieltjes integral for $\int _{-\infty }^{\infty }f(x)\,dH(x)$ where $H$ is the heaviside function. I think this should give $f(0)$ but when I want to imagine the $(g(t),f(t))$ curve, it starts by drawing a vertical line of height $\infty$ at $-\infty$ since $H(t)=0$ on $(-\infty,0)$ so I don't really understand how to get back $f(0)$.

@HaraldHanche-Olsen, thank you. Why do we need $f\ge 0$?

will $p(\pi_*((\frac{\partial H}{\partial p}, -\frac{\partial H}{\partial q})))$ be equal to the action?

@JoshBurby doesn't the Frechet derivative need the concept of norm?

@JoshBurby, very interesting, I am starting to learn about this stuff, would you have any good reading recommendations?