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 |
Parametrix method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parametrix_method&oldid=49514