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 $G$ 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
[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