Set of Malgrange Speedwell

The Malgrange-Ehrenpreis theorem is an existence theorem from the theory of partial differential equations . It ensures the existence of a Green function for linear partial differential operators with constant coefficients. The set was found independently by Bernard Malgrange and Leon Ehrenpreis in the mid-1950s .

Terms

A linear, partial differential operator with constant coefficients arises from a polynomial in indeterminate by inserting the partial derivative for the -th indeterminate . Is ${\displaystyle P}$ ${\displaystyle n}$ ${\displaystyle i}$ ${\displaystyle D_{i}={\tfrac {\partial }{\partial x_{i}}}}$

${\displaystyle P=P(x_{1},\ldots ,x_{n})=\sum _{j_{1},\ldots ,j_{n}=0}^{N}\beta _{j_{1},\ldots ,j_{n}}x_{1}^{j_{1}}\ldots x_{n}^{j_{n}}}$

with coefficients , where the upper summation limit is a fixed natural number, so is ${\displaystyle \beta _{j_{1},\ldots ,j_{n}}\in \mathbb {C} }$ ${\displaystyle N}$

${\displaystyle P(D)=P(D_{1},\ldots ,D_{n})=\sum _{j_{1},\ldots ,j_{n}=0}^{N}\beta _{j_{1},\ldots ,j_{n}}{\frac {\partial ^{j_{1}}}{\partial x_{1}^{j_{1}}}}\ldots {\frac {\partial ^{j_{n}}}{\partial x_{n}^{j_{n}}}}}$

and the partial differential equation

${\displaystyle P(D)u=f}$

if the right-hand side is given, it is called linear, partial differential equation with constant coefficients , since the coefficients are not functions of the variables, but constants. The wave equation and Poisson's equation are typical examples. ${\displaystyle f}$ ${\displaystyle \beta _{j_{1},\ldots ,j_{n}}}$

The above differential equation is not only useful for functions, but also for distributions . If you take the delta distribution as the right-hand side , a distribution solution of the equation is called a Green function, even if it is not a classic function. If there is any right hand side and the convolution can be formed, then there is a solution of because of the constant coefficients . ${\displaystyle G}$ ${\displaystyle f}$ ${\displaystyle G*f}$ ${\displaystyle u=G*f}$ ${\displaystyle P(D)u=f}$

Therefore the differential equation is considered to be solved when a Green function is found. This underlines the importance of the following sentence:

Formulation of the sentence

Malgrange Ehrenpreis theorem : Let be a linear, partial differential operator with constant coefficients. Then the corresponding partial differential equation has a Green function. ${\displaystyle P(D)}$

Remarks

The original evidence used Hahn-Banach's Theorem and was therefore non-constructive. Constructive evidence is now also known.

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

Individual evidence

1. ↑ Bernard Malgrange: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. , Ann. Inst. Fourier, Vol. 6 (1955-1956), pp. 271-355
2. ^ Leon Ehrenpreis: Solution of some problems of division. I. Division by a polynomial of derivation. , Amer. J. Math. 76, 883-903 (1954)
3. ^ Leon Ehrenpreis: Solution of some problems of division. II. Division by a punctual distribution. , Amer. J. Math., Vol. 77 (1955), pp. 286-292
4. ↑ Milan Miklavcic: Applied Functional Analysis and Partial Differential Equations , World Scientific Pub Co (1998), ISBN 981-02-3535-6 , Theorem 3.3.4
5. ^ Kosaku Yosida: Functional Analysis. Classics in Mathematics. , Springer-Verlag (1995), ISBN 3-540-58654-7 , Chapter VI, Section 10: The Malgrange-Ehrenpreis Theorem
6. ^ Peter Wagner: A new constructive proof of the Malgrange-Ehrenpreis theorem , Amer. Math. Monthly, Vol. 116 (2009), pp. 457-462