Malgrange–Ehrenpreis theorem

The set of Malgrange - Ehrenpreis is an existence theorem from the theory of partial differential equations. He ensures the existence of a Green's function for linear partial differential operators with constant coefficients. The rate was independent of Bernard Malgrange and Leon found the mid-1950s Speedwell.

Terms

A linear partial differential operator with constant coefficients arises from a polynomial in indeterminates by the partial derivative is used for th Non- voice. is

Where the upper summation limit is a fixed natural number, then

And PDE

For a given right-hand side is a linear partial differential equation with constant coefficients, since the coefficients are not constants but functions of the variables. The wave equation and the Poisson equation are typical examples.

The above differential equation is now not only useful for functions but also for distributions. Taking as a right side the delta function, this means a distribution solution of the equation is a Green's function, even if it is not a classical function. Now if some right side and one can form the convolution, it is because of the constant coefficients of a solution.

Therefore, the differential equation with the discovery of a Green's function as solved. This underlines the importance of the following sentence:

Formulation of the sentence

Set of Malgrange - Ehrenpreis: There is a linear partial differential operator with constant coefficients. Then the associated partial differential equation has a Green's function.

Comments

The original proof used the Hahn- Banach and were therefore non- constructive. Meanwhile, constructive proofs are known.

Obvious generalizations to linear partial differential equations with non- constant coefficients do not apply, as the example of Lewy occupied.