One of the methods for studying boundary value problems for differential equations with variable coefficients by means of integral equations.
Suppose that in some region of the -dimensional Euclidean space one considers an elliptic differential operator (cf. Elliptic partial differential equation) of order ,
In (1) the symbol is a multi-index, , where the are non-negative integers, , , . With every operator (1) there is associated the homogeneous elliptic operator
with constant coefficients, where is an arbitrary fixed point. Let denote a fundamental solution of depending parametrically on . Then the function is called the parametrix of the operator (1) with a singularity at .
In particular, for the second-order elliptic operator
one can take as parametrix with singularity at the Levi function
In (2), , is the determinant of the matrix ,
and are the elements of the matrix inverse to .
Let be the integral operator
acting on functions from and let
Since, by definition of a fundamental solution,
where is the identity operator, one has
This equality indicates that for every sufficiently-smooth function of compact support in there is a representation
then is a solution of the equation
Thus, the question of the local solvability of reduces to that of invertibility of .
If one applies to functions that vanish outside a ball of radius with centre at , then for a sufficiently small the norm of can be made smaller than one. Then the operator exists; consequently, also exists, which is the inverse operator to . Here is an integral operator with as kernel a fundamental solution of .
The name parametrix is sometimes given not only to the function , but also to the integral operator with the kernel , as defined by (3).
In the theory of pseudo-differential operators, instead of a parametrix of is defined as an operator such that and are integral operators with infinitely-differentiable kernels (cf. Pseudo-differential operator). If only (or ) is such an operator, then is called a left (or right) parametrix of . In other words, in (4) is a left parametrix if in this equality has an infinitely-differentiable kernel. If has a left parametrix and a right parametrix , then each of them is a parametrix. The existence of a parametrix has been proved for hypo-elliptic pseudo-differential operators (see ).
|||L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)|
|||C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)|
|||L. Hörmander, , Pseudo-differential operators , Moscow (1967) (In Russian; translated from English)|
The operator is called the principal part of , cf. Principal part of a differential operator. The parametrix method was anticipated in two fundamental papers by E.E. Levi [a1], [a2]. The same procedure is also applicable for constructing the fundamental solution of a parabolic equation with variable coefficients (see, e.g., [a3]).
|[a1]||E.E. Levi, "Sulle equazioni lineari alle derivate parziali totalmente ellittiche" Rend. R. Acc. Lincei, Classe Sci. (V) , 16 (1907)|
|[a2]||E.E. Levi, "Sulle equazioni lineari totalmente ellittiche alle derivate parziali" Rend. Circ. Mat. Palermo , 24 (1907) pp. 275–317|
|[a3]||A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)|
|[a4]||L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985) pp. Chapts. 7; 18|
Parametrix method. Sh.A. Alimov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Parametrix_method&oldid=16020