Parametrix method

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 ,

(1)

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

(2)

In (2), , is the determinant of the matrix ,

and are the elements of the matrix inverse to .

Let be the integral operator

(3)

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

(4)

Moreover, if

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 [3]).

References

[1]	L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)
[2]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
[3]	L. Hörmander, , Pseudo-differential operators , Moscow (1967) (In Russian; translated from English)

Comments

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]).

References

[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