View source for Parametrix method

<!--
p0715701.png
$#A+1 = 72 n = 0
$#C+1 = 72 : ~/encyclopedia/old_files/data/P071/P.0701570 Parametrix method
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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  $  G $
of the  $  n $-
dimensional Euclidean space  $  \mathbf R  ^ {n} $
one considers an elliptic differential operator (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]) of order  $  m $,

$$ \tag{1 }
L( x, D)  =  \sum _ {| \alpha | \leq  m } a _  \alpha  ( x) D  ^  \alpha  .
$$

In (1) the symbol  $  \alpha $
is a multi-index,  $  \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $, 
where the  $  \alpha _ {j} $
are non-negative integers,  $  | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} $, 
$  D  ^  \alpha  = D _ {1} ^ {\alpha _ {1} } \dots D _ {n} ^ {\alpha _ {n} } $, 
$  D _ {j} = - i \partial  / \partial  x _ {j} $. 
With every operator (1) there is associated the homogeneous elliptic operator

$$ 
L _ {0} ( x _ {0} , D)  =  \sum _ {| \alpha | = m } a _  \alpha  ( x _ {0} ) D
 ^  \alpha 
$$

with constant coefficients, where  $  x _ {0} \in G $
is an arbitrary fixed point. Let  $  \epsilon ( x, x _ {0} ) $
denote a [[Fundamental solution|fundamental solution]] of  $  L _ {0} ( x _ {0} , D) $
depending parametrically on  $  x _ {0} $. 
Then the function  $  \epsilon ( x , x _ {0} ) $
is called the parametrix of the operator (1) with a singularity at  $  x _ {0} $.

In particular, for the second-order elliptic operator

$$ 
L( x, D)  =  \sum _ {i,j= 1 } ^ { n }  a _ {ij} ( x) 
\frac{\partial   ^ {2} }{\partial 
x _ {i} \partial  x _ {j} }
 + \sum _ { i= } 1 ^ { n }  b _ {i} ( x) 
\frac \partial {\partial 
x _ {i} }
 + c ( x)
$$

one can take as parametrix with singularity at  $  y $
the Levi function

$$ \tag{2 }
\epsilon ( x, y)  =  \left \{

In (2),  $  \omega _ {n} = 2 \pi  ^ {n/2} / \Gamma ( n/2) $, 
$  A( y) $
is the determinant of the matrix  $  \| \alpha _ {ij} ( y) \| $,

$$ 
R( x, y)  =  \sum _ { i,j= } 1 ^ { {n }  } A _ {ij} ( y)( x _ {i} - y _ {i} )( x _ {j} - y _ {j} ),
$$

and  $  A _ {ij} ( y) $
are the elements of the matrix inverse to  $  \| \alpha _ {ij} ( y) \| $.

Let  $  S _ {x _ {0}  } $
be the integral operator

$$ \tag{3 }
( S _ {x _ {0}  } \phi )( x)  =  \int\limits _ { G } \epsilon ( x- y, x _ {0} ) \phi ( y)  dy ,
$$

acting on functions from  $  C _ {0}  ^  \infty  ( G) $
and let

$$ 
T _ {x _ {0}  }  =  S _ {x _ {0}  } [ L _ {0} ( x _ {0} , D) - L( x, D)] .
$$

Since, by definition of a fundamental solution,

$$ 
L _ {0} ( x _ {0} , D) S _ {x _ {0}  }  =  S _ {x _ {0}  } L _ {0} ( x _ {0} , D)
 =  I,
$$

where  $  I $
is the identity operator, one has

$$ 
I  =  S _ {x _ {0}  } L( x, D) + T _ {x _ {0}  } .
$$

This equality indicates that for every sufficiently-smooth function  $  \phi $
of compact support in  $  G $
there is a representation

$$ \tag{4 }
\phi  =  S _ {x _ {0}  } L ( x, D) \phi + T _ {x _ {0}  } \phi .
$$

Moreover, if

$$ 
\phi  =  S _ {x _ {0}  } f + T _ {x _ {0}  } \phi ,
$$

then  $  \phi $
is a solution of the equation

$$ 
L( x, D) \phi  =  f.
$$

Thus, the question of the local solvability of  $  L _  \phi  = f $
reduces to that of invertibility of  $  I- T _ {x _ {0}  } $.

If one applies  $  T _ {x _ {0}  } $
to functions  $  \phi $
that vanish outside a ball of radius  $  R $
with centre at  $  x _ {0} $, 
then for a sufficiently small  $  R $
the norm of  $  T _ {x _ {0}  } $
can be made smaller than one. Then the operator  $  ( I- T _ {x _ {0}  } )  ^ {-} 1 $
exists; consequently, also  $  E = ( I- T _ {x _ {0}  } )  ^ {-} 1 S _ {x _ {0}  } $
exists, which is the inverse operator to  $  L( x, D) $. 
Here  $  E $
is an integral operator with as kernel a fundamental solution of  $  L( x, D) $.

The name parametrix is sometimes given not only to the function  $  \epsilon ( x, x _ {0} ) $, 
but also to the integral operator  $  S _ {x _ {0}  } $
with the kernel  $  \epsilon ( x, x _ {0} ) $, 
as defined by (3).

In the theory of pseudo-differential operators, instead of  $  S _ {x _ {0}  } $
a parametrix of  $  L( x, D) $
is defined as an operator  $  S $
such that  $  I- L( x, D) S $
and  $  I- SL( x, D) $
are integral operators with infinitely-differentiable kernels (cf. [[Pseudo-differential operator|Pseudo-differential operator]]). If only  $  I- SL $(
or  $  I- LS $) 
is such an operator, then  $  S $
is called a left (or right) parametrix of  $  L( x, D) $. 
In other words,  $  S _ {x _ {0}  } $
in (4) is a left parametrix if  $  T _ {x _ {0}  } $
in this equality has an infinitely-differentiable kernel. If  $  L( x, D) $
has a left parametrix  $  S  ^  \prime  $
and a right parametrix  $  S  ^ {\prime\prime} $, 
then each of them is a parametrix. The existence of a parametrix has been proved for hypo-elliptic pseudo-differential operators (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bers,   F. John,   M. Schechter,   "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Miranda,   "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Hörmander,  , ''Pseudo-differential operators'' , Moscow  (1967)  (In Russian; translated from English)</TD></TR></table>

====Comments====
The operator  $  L _ {0} ( x, D) $
is called the principal part of  $  L $, 
cf. [[Principal part of a differential operator|Principal part of a differential operator]]. The parametrix method was anticipated in two fundamental papers by E.E. Levi [[#References|[a1]]], [[#References|[a2]]]. The same procedure is also applicable for constructing the fundamental solution of a parabolic equation with variable coefficients (see, e.g., [[#References|[a3]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.E. Levi,   "Sulle equazioni lineari alle derivate parziali totalmente ellittiche"  ''Rend. R. Acc. Lincei, Classe Sci. (V)'' , '''16'''  (1907)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.E. Levi,   "Sulle equazioni lineari totalmente ellittiche alle derivate parziali"  ''Rend. Circ. Mat. Palermo'' , '''24'''  (1907)  pp. 275–317</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1–4''' , Springer  (1983–1985)  pp. Chapts. 7; 18</TD></TR></table>