the Malgrange-Ehrenpreis theorem
/ mal-GRAHNZH EH-ren-price /
A fundamental solution of a differential operator is its response to a point source — the single building block from which solutions to any source are assembled by superposition. A natural worry is whether such a building block always exists. The Malgrange-Ehrenpreis theorem answers this with a clean and powerful yes, for a wide and important class of operators.
Precisely, the theorem states that every nonzero linear partial differential operator with constant coefficients has a fundamental solution: there exists a distribution E such that L applied to E equals the Dirac delta. The proof lives squarely in distribution theory and Fourier analysis — one looks for E by dividing by the symbol of the operator in frequency space and handling its zeros carefully. The conclusion is that constant-coefficient operators are never obstructed: no matter how exotic such an L is, a point-source response exists as a distribution, even when it could not exist as a classical function. Once you have E, the equation L u = f is solved by convolution, u = E convolved with f, whenever that convolution is defined.
This matters as the foundational guarantee underneath transform and Green's-function methods for constant-coefficient PDEs. It tells you the whole approach is never empty: the object you are hunting for always exists. The crucial fine print is the hypothesis: constant coefficients. For variable-coefficient operators the statement can fail — there are famous examples (due to Hans Lewy) of smooth operators with no solution at all near a point — so existence is genuinely special, not automatic.
The Laplacian in three dimensions has the fundamental solution E(x) = -1/(4 pi |x|): one can check that Laplacian E = delta in the distributional sense. Malgrange-Ehrenpreis guarantees that some such E exists for every constant-coefficient operator, even ones with no elementary formula.
Every constant-coefficient operator has a point-source response — existence is guaranteed even without an explicit formula.
The guarantee needs constant coefficients. For variable coefficients it can fail outright: Lewy's example is a first-order operator with smooth coefficients having no solution near a point — so do not assume a fundamental solution always exists.