# Reference request: Systems of linear PDES with constant coefficients

I am looking for a reference for the following statement:

Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs \begin{align} P_i(\partial / \partial x_1, \dots, \partial / \partial x_m)u = f_i, \quad i = 1, \dots, k \end{align} where $u(x_1, \dots, x_m)$ is a scalar function to be found.

Then, if this system has a solution, we should obviously have $$\sum_{i=1}^k Q_i(\partial / \partial x_1, \dots, \partial / \partial x_m)f_i = 0$$ for each $k$-tuple $(Q_1, \dots, Q_k)$ such that $$\sum_{i=1}^k Q_iP_i = 0.$$ The statement is that this condition is also sufficient for local solvability of our system of PDEs. More precisely,

1) if $f_1, \dots, f_k$ are $\mathrm C^\infty$ functions in a neighborhood of $x \in \mathbb R^m$, then there exists a $\mathrm C^\infty$ solution in a (possibly smaller) neighborhood of $x$;

2) if $f_1, \dots, f_k$ are analytic, then the solution is also analytic.

I suspect that this should be Ehrenpreis or Malgrange, but I was not able to find the precise statement.

Another question is whether there is a geometric way to understand this result. For instance, if $P_1, \dots, P_k$ are of degree $1$, then the statement can be deduced from the Frobenius theorem.

• In your conclusion (2), I think you mean to say that 'if $f_1,\ldots,f_k$ are analytic then there exists a solution $u$ that is analytic'. (Obvious examples show that not all of the solutions need be analytic, even when the $f_i$ are all zero.) Such a version of the statement (at least for local solutions) follows from the Cartan-Kähler Theorem. Nov 11, 2014 at 18:33
• Yes, of course, I meant that there exists a solution. Nov 11, 2014 at 23:28

I think what you are looking for is something like Theorem 9 in the following reference: Ehrenpreis, Leon, A fundamental principle for systems of linear differential equations with constant coefficients, and some of its applications. (1961) Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) pp. 161–174 Jerusalem Academic Press, Jerusalem; Pergamon, Oxford.

• Thank you, Robert, that's it. I also found nice exposition of this result in the book Fourier analysis in several complex variables'' by Ehrenpreis. Nov 11, 2014 at 23:31
• @Robert Bryant Dear prof. Bryant, are there similar results to the Ehrenpreis Fundamental Principle if the coefficients are instead all very close to constants? Could you perhaps provide a reference in the case? Thank you for your patience.
– user17697
Sep 21, 2016 at 10:55

Another reference for results on solvability of linear constant coefficient PDEs is

Tarkhanov, N. N. Complexes of Differential Operators, vol. 340 of Mathematics and Its Applications (Kluwer, Dordrecht, 1995) http://dx.doi.org/10.1007/978-94-011-0327-5

In particular, see Secs. 1.2.2 and 1.2.4. The latter comes with extensive references on this subject. The works of Ehrenpreis, Malgrange and Palamodov get a particular mention, with final credit given to Malgrange.

Theorem 7.6.13 in Hörmander's book "An Introduction to Complex Analysis in Several Variables" (North-Holland, 1990, 3rd edition) contains the result you are looking for. Sections 7.6 and 7.7 of this book give a full prove of the Ehrenpreis Fundamental Principle of which solvability is a part. See Hörmander's notes at the end of chapter 7 for historical information.