Parametrix

A parametrix is an object from the mathematical branch of the theory of partial differential equations. It is particularly used in the theory of partial differential equations and is a generalization of the fundamental solution of a differential operator with constant coefficients.

definition

A fundamental solution of a differential operator with constant coefficients is a distribution such that ${\ displaystyle P (D)}$ ${\ displaystyle u}$

{\ displaystyle P (D) u = \ delta}

in the (distributional sense) applies. The symbol here denotes the delta distribution . ${\ displaystyle \ delta}$

A parametrix of the differential operator with constant coefficients is a distribution such that ${\ displaystyle P (D)}$ ${\ displaystyle u \ in {\ mathcal {D}} (\ mathbb {R} ^ {n})}$

{\ displaystyle P (D) u = \ delta + \ omega}

holds, where is a smooth function. ${\ displaystyle \ omega \ in C ^ {\ infty} (\ mathbb {R} ^ {n})}$

In particular, the fundamental solution is a special case of Parametrix. The Parametrix is a useful concept for studying elliptic differential operators .

Pseudo differential operators

In the theory of (hypo) elliptic pseudo differential operators , the term parametrix is used somewhat differently.

So be an actually carried pseudo differential operator of the order . Then a pseudo differential operator of order Parametrix is called , if ${\ displaystyle P}$ ${\ displaystyle k}$ ${\ displaystyle Q}$ ${\ displaystyle -k}$ ${\ displaystyle P}$

{\ displaystyle P \ circ Q = I + R_ {1} \ quad {\ text {and}} \ \ quad Q \ circ P = I + R_ {2}}

applies. It is the identity operator and and are smoothing pseudo-differential operators, that is, they have the order . ${\ displaystyle I}$ ${\ displaystyle R_ {1}}$ ${\ displaystyle R_ {2}}$ ${\ displaystyle - \ infty}$

literature

Lars Hörmander : The analysis of linear partial differential operators I Grundl. Math. Science. Vol. 256, Springer, 1983, ISBN 3-540-12104-8
Sh.A. Alimov: Parametrix method . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).