I would assume, for example, there's a generalization of the Euler-Lagrange (E-L) equations for operators rather than functionals. The E-L normally lead to a system of differential equations. In the more general case maybe they lead to a family of differential operators? but It's not obvious to me.

Thank you for checking. I think what I've found interesting in these computations is that characterization in terms of family of measures. I wonder therefore if there's a general theory that has a bit of focus on these problems. Or if a numerical method can be derived to find $f$ given such conditions.

And I guess the way to prove this is by solving the PDE with weak solution and show that those extra terms don't contribute to the solution

So in practice there's no singularity (case $n =2,3$) because of the "distribution sense", right?

The opinion is fine and I respect it, as proved by the fact I've opened this question based on your suggestion and I hope you appreciate that. Discouraging comments are not fine, the rest of your answer covers many points that are constructive, which I am not disputing. You can have your opinion obviously but phrase it less harshly. This is a public platform and I want as many people as possible to jump and commenting constructively. A remark like yours probably does not help with that.

Thank you for your detailed answer. I would however discouraging comments like "approach is doomed to fail". I was trying to look at the problem, simple maybe, from another perspective to gain some extra insights. So to answer your remark (even if you didn't ask) is "yes, I was overcomplicating the problem on purpose".

@WillieWong In the body of the question I've defined $u(t)$.

See this: mathoverflow.net/questions/467299/…

As I am assuming $H$ is finite sum of linear combination of translation operators I think the closeness is there. Am I wrong?

What about the case where $H$ is linear combination of translation operators?

How did you get that $Hx = y$ is implied by $\left\langle Ex, Hx - y \right\rangle = 0$, I think $H(h)$ is eventually "a solution" not "all solutions".

I changed the question to have 2. as addition