# Parametrix method

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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)