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Let $L$ be any differential operator (not necessarily linear).

Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form:

Given a boundary value problem on some domain $\Omega$. If $L$, the initial condition, and the boundary are of a certain form, then separation of variables will yield the solution to the boundary value problem.

Do such general statements exist, and if they do, where can I find them? I am interested in statements about strong (classical) solutions and weak (distributional) solutions.

I am aware that questions similar to this one are already posted on this site, however, all of them consider specific differential operators and then they focus on separability of the domain. I am interested in statements about general differential operators.

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The short answer is No. A major problem is that there is no single universal definition for what it means for an equation to be solvable by separation of variables. This defect in the theory, as well as the lack of general statements of the form that you would like to see, was highlighted a while ago in this BAMS book review:

Tom H. Koornwinder. "Review: Willard Miller, Jr., Symmetry and separation of variables." Bull. Amer. Math. Soc. (N.S.) 1 (6) 1014 - 1019, November 1979.

Unfortunately, the situation has not changed so much since then. Koornwinder himself proposed a general definition for an equation to be separable and you can find a few papers that refer to it or extend it by looking up papers that cite this review. But there are no really general results, as far as I am aware.

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  • $\begingroup$ I was suspecting that this would be the case, but I was hoping to be wrong. Thanks for your answer, I'll see if I can find anything in the papers that cite the review you linked. I'll leave the question open for a while so that other people can add to it if they want to, but otherwise I'll accept you answer tomorrow $\endgroup$ Nov 10, 2021 at 22:25
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    $\begingroup$ one thing to maybe keep in mind with separation of variables (or any pde technique that gives you a formula in the end). You can always be sloppy and once you have a formula then you can try and be rigourous and prove things... $\endgroup$
    – Math604
    Nov 11, 2021 at 2:08
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    $\begingroup$ The comment of @Math604 expresses a general strategy in mathematics. To prove something exists with such-and-such a property: search for it; look for it; dig for it; hunt for it... it you're lucky (or if you're a good hunter) then you'll find it, you'll express it precisely, and then you'll prove rigorously that it satisfies the required property. $\endgroup$
    – Lee Mosher
    Nov 11, 2021 at 12:24

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