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.

The original proofs of Malgrange and Ehrenpreis did not use explicit constructions as they used the Hahn–Banach theorem. Since then several constructive proofs have been found.

There is a very short proof using the Fourier transform and the Bernstein–Sato polynomial, as follows. By taking Fourier transforms the Malgrange–Ehrenpreis theorem is equivalent to the fact that every non-zero polynomial ${\displaystyle P}$ has a distributional inverse. By replacing ${\displaystyle P}$ by the product with its complex conjugate, one can also assume that ${\displaystyle P}$ is non-negative. For non-negative polynomials ${\displaystyle P}$ the existence of a distributional inverse follows from the existence of the Bernstein–Sato polynomial, which implies that ${\displaystyle P^{s}}$ can be analytically continued as a meromorphic distribution-valued function of the complex variable ${\displaystyle s}$; the constant term of the Laurent expansion of ${\displaystyle P^{s}}$ at ${\displaystyle s=-1}$ is then a distributional inverse of ${\displaystyle P}$.

Other proofs, often giving better bounds on the growth of a solution, are given in (Hörmander 1983a, Theorem 7.3.10), (Reed & Simon 1975, Theorem IX.23, p. 48) and (Rosay 1991). (Hörmander 1983b, chapter 10) gives a detailed discussion of the regularity properties of the fundamental solutions.

A short constructive proof was presented in (Wagner 2009, Proposition 1, p. 458):

${\displaystyle E={\frac {1}{\overline {P_{m}(2\eta )}}}\sum _{j=0}^{m}a_{j}e^{\lambda _{j}\eta x}{\mathcal {F}}_{\xi }^{-1}\left({\frac {\overline {P(i\xi +\lambda _{j}\eta )}}{P(i\xi +\lambda _{j}\eta )}}\right)}$

is a fundamental solution of ${\displaystyle P(\partial )}$, i.e., ${\displaystyle P(\partial )E=\delta }$, if ${\displaystyle P_{m}}$ is the principal part of ${\displaystyle P}$, ${\displaystyle \eta \in \mathbb {R} ^{n}}$ with ${\displaystyle P_{m}(\eta )\neq 0}$, the real numbers ${\displaystyle \lambda _{0},\ldots ,\lambda _{m}}$ are pairwise different, and

${\displaystyle a_{j}=\prod _{k=0,k\neq j}^{m}(\lambda _{j}-\lambda _{k})^{-1}.}$

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