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One of the methods for studying boundary value problems for differential equations with variable coefficients by means of integral equations.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715701.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715702.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715703.png" /> one considers an elliptic differential operator (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715704.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
L( x, D)  = \sum _ {| \alpha | \leq  m } a _  \alpha  ( x) D  ^  \alpha  .
 +
$$
  
In (1) the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715706.png" /> is a multi-index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715707.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715708.png" /> are non-negative integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p0715709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157011.png" />. With every operator (1) there is associated the homogeneous elliptic operator
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157012.png" /></td> </tr></table>
+
$$
 +
L _ {0} ( x _ {0} , D)  = \sum _ {| \alpha | = m } a _  \alpha  ( x _ {0} ) D
 +
^  \alpha
 +
$$
  
with constant coefficients, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157013.png" /> is an arbitrary fixed point. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157014.png" /> denote a [[Fundamental solution|fundamental solution]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157015.png" /> depending parametrically on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157016.png" />. Then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157017.png" /> is called the parametrix of the operator (1) with a singularity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157018.png" />.
+
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
 
In particular, for the second-order elliptic operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157019.png" /></td> </tr></table>
+
$$
 +
L( x, D)  = \sum _ {i,j= 1 } ^ { n }  a _ {ij} ( x)
 +
\frac{\partial  ^ {2} }{\partial
 +
x _ {i} \partial  x _ {j} }
 +
+ \sum _ { i= } 1 ^ { n }  b _ {i} ( x)
 +
\frac \partial {\partial
 +
x _ {i} }
 +
+ c ( x)
 +
$$
 +
 
 +
one can take as parametrix with singularity at  $  y $
 +
the Levi function
 +
 
 +
$$ \tag{2 }
 +
\epsilon ( x, y)  = \left \{
 +
 
 +
\begin{array}{ll}
 +
 
 +
\frac{1}{( n- 2) \omega _ {n} \sqrt {A( y) } }
 +
[ R( x, y)]  ^ {2-} n ,  & n > 2,  \\
  
one can take as parametrix with singularity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157020.png" /> the Levi function
+
\frac{1}{2 \pi \sqrt A( y) }
 +
  \mathop{\rm ln}  R( x, y) ,  & n = 2 . \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\right .$$
  
In (2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157023.png" /> is the determinant of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157024.png" />,
+
In (2), $  \omega _ {n} = 2 \pi  ^ {n/2} / \Gamma ( n/2) $,  
 +
$  A( y) $
 +
is the determinant of the matrix $  \| \alpha _ {ij} ( y) \| $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157025.png" /></td> </tr></table>
+
$$
 +
R( x, y)  = \sum _ { i,j= } 1 ^ { {n }  } A _ {ij} ( y)( x _ {i} - y _ {i} )( x _ {j} - y _ {j} ),
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157026.png" /> are the elements of the matrix inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157027.png" />.
+
and $  A _ {ij} ( y) $
 +
are the elements of the matrix inverse to $  \| \alpha _ {ij} ( y) \| $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157028.png" /> be the integral operator
+
Let $  S _ {x _ {0}  } $
 +
be the integral operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
( S _ {x _ {0}  } \phi )( x)  = \int\limits _ { G } \epsilon ( x- y, x _ {0} ) \phi ( y) dy ,
 +
$$
  
acting on functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157030.png" /> and let
+
acting on functions from $  C _ {0}  ^  \infty  ( G) $
 +
and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157031.png" /></td> </tr></table>
+
$$
 +
T _ {x _ {0}  }  = S _ {x _ {0}  } [ L _ {0} ( x _ {0} , D) - L( x, D)] .
 +
$$
  
 
Since, by definition of a fundamental solution,
 
Since, by definition of a fundamental solution,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157032.png" /></td> </tr></table>
+
$$
 +
L _ {0} ( x _ {0} , D) S _ {x _ {0}  }  = S _ {x _ {0}  } L _ {0} ( x _ {0} , D)
 +
= I,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157033.png" /> is the identity operator, one has
+
where $  I $
 +
is the identity operator, one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157034.png" /></td> </tr></table>
+
$$
 +
= S _ {x _ {0}  } L( x, D) + T _ {x _ {0}  } .
 +
$$
  
This equality indicates that for every sufficiently-smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157035.png" /> of compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157036.png" /> there is a representation
+
This equality indicates that for every sufficiently-smooth function $  \phi $
 +
of compact support in $  G $
 +
there is a representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\phi  = S _ {x _ {0}  } L ( x, D) \phi + T _ {x _ {0}  } \phi .
 +
$$
  
 
Moreover, if
 
Moreover, if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157038.png" /></td> </tr></table>
+
$$
 +
\phi  = S _ {x _ {0}  } f + T _ {x _ {0}  } \phi ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157039.png" /> is a solution of the equation
+
then $  \phi $
 +
is a solution of the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157040.png" /></td> </tr></table>
+
$$
 +
L( x, D) \phi  = f.
 +
$$
  
Thus, the question of the local solvability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157041.png" /> reduces to that of invertibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157042.png" />.
+
Thus, the question of the local solvability of $  L _  \phi  = f $
 +
reduces to that of invertibility of $  I- T _ {x _ {0}  } $.
  
If one applies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157043.png" /> to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157044.png" /> that vanish outside a ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157045.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157046.png" />, then for a sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157047.png" /> the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157048.png" /> can be made smaller than one. Then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157049.png" /> exists; consequently, also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157050.png" /> exists, which is the inverse operator to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157051.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157052.png" /> is an integral operator with as kernel a fundamental solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157053.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157054.png" />, but also to the integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157055.png" /> with the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157056.png" />, as defined by (3).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157057.png" /> a parametrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157058.png" /> is defined as an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157061.png" /> are integral operators with infinitely-differentiable kernels (cf. [[Pseudo-differential operator|Pseudo-differential operator]]). If only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157062.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157063.png" />) is such an operator, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157064.png" /> is called a left (or right) parametrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157065.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157066.png" /> in (4) is a left parametrix if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157067.png" /> in this equality has an infinitely-differentiable kernel. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157068.png" /> has a left parametrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157069.png" /> and a right parametrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157070.png" />, then each of them is a parametrix. The existence of a parametrix has been proved for hypo-elliptic pseudo-differential operators (see [[#References|[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====
 
====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>
 
<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====
 
====Comments====
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157071.png" /> is called the principal part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071570/p07157072.png" />, 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]]]).
+
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====
 
====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>
+
<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 {{ZBL|38.0402.01}}</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>

Latest revision as of 18:05, 22 May 2024


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) 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 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

$$ \tag{2 } \epsilon ( x, y) = \left \{ \begin{array}{ll} \frac{1}{( n- 2) \omega _ {n} \sqrt {A( y) } } [ R( x, y)] ^ {2-} n , & n > 2, \\ \frac{1}{2 \pi \sqrt A( y) } \mathop{\rm ln} R( x, y) , & n = 2 . \\ \end{array} \right .$$

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). 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 [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 $ L _ {0} ( x, D) $ is called the principal part of $ L $, 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 Zbl 38.0402.01
[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
How to Cite This Entry:
Parametrix method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parametrix_method&oldid=49355
This article was adapted from an original article by Sh.A. Alimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article