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