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Having an energy functional one can write the Euler-Lagrange equations. I want to do the reverse. Having a system of PDEs, I want to find an energy functional that "best" matches those. It is clear that not every system of PDEs is a system of Euler-Lagrange equations, and in the specific cases I am considering this needs not to be the case. Nevertheless I would like to find a functional whose solutions best match the solutions from my system of PDEs (wrt some metric - for example L2). Are there any references on this topic?

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    $\begingroup$ I looked for exactly the same thing about a year ago. I was looking to derive an set of energy functionals for a system of reduced PDEs that I had derived. Alas, the only concrete thing I ever found was this, which is not much to go on. $\endgroup$ Jun 19, 2020 at 14:47
  • $\begingroup$ @mattos Yeah, I was thinking about something more similar to minimizing a functional. What you linked is specific to FEM and simply rewrites the equations in their weak form. $\endgroup$
    – lightxbulb
    Jun 19, 2020 at 14:54
  • $\begingroup$ I think this Q&A would give you some nice references for your problem: the topics you are investigating is called "inverse problem of the calculus of variation", and I advice you also to have a look at the web site of Enzo Tonti (linked in the Q&A above) in order to have a quick introduction to his general method of solving the problem. $\endgroup$ Jul 3, 2020 at 5:43
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    $\begingroup$ @DanieleTampieri The paper from Tonti is currently a bit over my head, but this resource was nevertheless the most helpful thus far. Feel free to make your comment an answer so I cam accept it. $\endgroup$
    – lightxbulb
    Jul 3, 2020 at 12:40

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I think this Q&A gives you some nice references for your problem: the topic you are investigating is called "inverse problem of the calculus of variation", and I advice you also to have a look at the web site of Enzo Tonti. In particular, recapitulating briefly, Tonti's main result (in its more recent formulation [1]), given a general (linear/nonlinear, local/nonlocal etc.) operator equation $$ \mathsf{N}\big(x,u(x)\big)=f(x)\label{1}\tag{1} $$ one can always find a functional $F$ so that its functional derivative $$ \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon}F(u+\varepsilon\varphi)\right|_{\varepsilon=0}=0\;\text{ for all $\varphi \in$ domain of $\mathsf{N}$}\iff u\text{ is a solution of \eqref{1}}\label{2}\tag{2} $$ Moreover, $F$ can be chosen in such a way that $u$ is a minimum for it. However, it is worth to remember that the equation(s) resulting from imposing the vanishing of the functional derivative of $F$ may have not always the form of Euler-Lagrange equations: it depends on the structure of \eqref{1} if this happens or not (see the reference by Filippov in the above linked Q&A for more details on such matters).

Reference

[1] Tonti, Enzo, "Extended variational formulation", Vestnik Rossiĭskogo Universiteta Druzhby Narodov, Seriya Matematika 2, No. 2, 148-162 (1995). ZBL0965.35036.

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  • $\begingroup$ Are you aware of a text that goes into more practical details regarding the procedure described in Tonti's paper? Namely, more examples with different kernels $k$ for different PDEs, as well as examples with the degenerate kernels. A notable practical disadvantage is the mentioned doubling of the integrals. $\endgroup$
    – lightxbulb
    Jul 3, 2020 at 14:18
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    $\begingroup$ @lightxbulb I have to check in my library: I advice also to write to Enzo Tonti, as he is very kind and helpful. $\endgroup$ Jul 3, 2020 at 14:24
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    $\begingroup$ @lightxbulb I checked my sources, and it seems that Enzo Tonti did not published later papers on such topic: also Filippov, his school and followers seem to devote their efforts to the investigation of theoretical side of it. On the use of degenerate kernels, you may consult the woks of Antonio Tralli cited at the end of Tonti's inked paper: though those techniques seem more commonly used in numerical computations on specific problems than on the theoretical side. And again I advise you to write to Enzo Tonti: that particular topic he may know other, more specific, researches. $\endgroup$ Jul 5, 2020 at 6:01
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    $\begingroup$ Thank you. I am currently reading the earlier works of Tonti to find more details on the subject. There seem to be various other techniques that allow for the initial problem formulation to be kept as differential equations and not turn them into integro-differential equations, but those do not work for every system of equations. I will check out Tralli's work too, thank you. I will most likely contact Tonti if I cannot find a reasonable solution after going through the literature. $\endgroup$
    – lightxbulb
    Jul 5, 2020 at 7:44
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    $\begingroup$ @lightxbulb well done. Since you are reading also earlier literature, it may be interesting for you to have a look at a large survey paper written by Filippov, Savchin and Shorokhov. Finally, good luck and success for your researches. $\endgroup$ Jul 5, 2020 at 8:01

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