Malgrange–Ehrenpreis theorem

A key question in mathematics and physics is how to model empty space with a point source, like the effect of a point mass on the gravitational potential energy, or a point heat source on a plate. Such physical phenomena are modeled by partial differential equations, having the form ${\displaystyle L\phi =\delta }$, where ${\displaystyle L}$ is a linear differential operator and ${\displaystyle \delta }$ is a delta function representing the point source. A solution to this problem (with suitable boundary conditions) is called a Green's function.

This motivates the question: given a linear differential operator ${\displaystyle L}$ (with constant coefficients), can we always solve ${\displaystyle L\phi =\delta }$? The Malgrange–Ehrenpreis theorem answers this in the affirmative. It states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956).

This means that the differential equation

${\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=\delta (\mathbf {x} ),}$

where ${\displaystyle P}$ is a polynomial in several variables and ${\displaystyle \delta }$ is the Dirac delta function, has a distributional solution ${\displaystyle u}$. It can be used to show that

${\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=f(\mathbf {x} )}$

has a solution for any compactly supported distribution ${\displaystyle f}$. The solution is not unique in general.

The analogue for differential operators whose coefficients are polynomials (rather than constants) is false: see Lewy's example.