Parametrix

It is useful to review what a fundamental solution for a differential operator P(D) with constant coefficients is: it is a distribution u on ℝⁿ such that

${\displaystyle P(D){u(x)}=\delta (x)~,}$

in the weak sense, where δ is the Dirac delta distribution.

In a similar way, a parametrix for a variable coefficient differential operator P(x,D) is a distribution u such that

${\displaystyle P(x,D){u(x)}=\delta (x)+\omega (x)~,}$

where ω is some C ^∞ function with compact support.

The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators with variable coefficient, since for such operators over appropriate domains a parametrix can be shown to exist, can be somewhat easily constructed[1] and be a smooth function away from the origin.[2]

Having found the analytic expression of the parametrix, it is possible to compute the solution of the associated fairly general elliptic partial differential equation by solving an associated Fredholm integral equation: also, the structure itself of the parametrix reveals properties of the solution of the problem without even calculating it, like its smoothness[3] and other qualitative properties.

More generally, if L is any pseudodifferential operator of order p, then another pseudodifferential operator L⁺ of order –p is called a parametrix for L if the operators

${\displaystyle L\circ L^{+}-I,\quad L^{+}\circ L-I}$

are both pseudodifferential operators of negative order. The operators L and L⁺ will admit continuous extensions to maps between the Sobolev spaces H^s and H^s+k.

On a compact manifold, the differences above are compact operators. In this case the original operator L defines a Fredholm operator between the Sobolev spaces.[4]

An explicit construction of a parametrix for second order partial differential operators based on power series developments was discovered by Jacques Hadamard. It can be applied to the Laplace operator, the wave equation and the heat equation.

In the case of the heat equation or the wave equation, where there is a distinguished time parameter t, Hadamard's method consists in taking the fundamental solution of the constant coefficient differential operator obtained freezing the coefficients at a fixed point and seeking a general solution as a product of this solution, as the point varies, by a formal power series in t. The constant term is 1 and the higher coefficients are functions determined recursively as integrals in a single variable.

In general, the power series will not converge but will provide only an asymptotic expansion of the exact solution. A suitable truncation of the power series then yields a parametrix.[5][6]

A sufficiently good parametrix can often be used to construct an exact fundamental solution by a convergent iterative procedure as follows (Berger, Gauduchon & Mazet 1971).

If L is an element of a ring with multiplication * such that

${\displaystyle L*P=1+R}$

for some approximate right inverse P and "sufficiently small" remainder term R then, at least formally,

${\displaystyle L*P*(1-R+R*R-R*R*R+\cdots )=1}$

so if the infinite series makes sense then L has a right inverse

${\displaystyle P-P*R+P*R*R-P*R*R*R+\cdots }$.

If L is a pseudo-differential operator and P is a parametrix, this gives a right inverse to L, in other words a fundamental solution, provided that R is "small enough" which in practice means that it should be a sufficiently good smoothing operator.

If P and R are represented by functions, then the multiplication * of pseudo-differential operators corresponds to convolution of functions, so the terms of the infinite sum giving the fundamental solution of L involve convolution of P with copies of R.

• Bejancu, A. (2001) [1994], "Parametrix method", Encyclopedia of Mathematics, EMS Press
• Berger, Marcel; Gauduchon, Paul; Mazet, Edmond (1971), Le spectre d'une variété riemannienne, Lecture Notes in Mathematics (in French), vol. 194, Berlin, New York: Springer-Verlag, pp. VII, 251, doi:10.1007/BFb0064643, ISBN 978-3-540-05437-5, MR 0282313, Zbl 0223.53034
• Hadamard, Jacques (2003) [1923], Lectures on Cauchy's problem in linear partial differential equations, Dover Phoenix editions, New York: Dover Publications, ISBN 978-0-486-49549-1, JFM 49.0725.04, MR 0051411, Zbl 0049.34805
• Hadamard, J. (1932), Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques (in French), Paris: Herman, JFM 58.0519.16, Zbl 0006.20501.
• Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundlehren der Mathematischen Wissenschaft, vol. 256, Heidelberg – Berlin – New York: Springer Verlag, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035, Zbl 0521.35001.
• Hörmander, L. (1985), The analysis of linear partial differential operators III, Grundlehren der Mathematischen Wissenschaft, vol. 274, Heidelberg – Berlin – New York: Springer Verlag, ISBN 3-540-13828-5, MR 0781536, Zbl 0601.35001.
• Levi, Eugenio Elia (1907), "Sulle equazioni lineari alle derivate parziali totalmente ellittiche", Rendiconti della Reale Accademia dei Lincei, Classe di Scienze Fisiche, Matematiche, Naturali, Serie V, 16 (12): 932–938, JFM 38.0403.01 (in Italian).
• Levi, Eugenio Elia (1907), "Sulle equazioni lineari totalmente ellittiche alle derivate parziali", Rendiconti del Circolo Matematico di Palermo, 24 (1): 275–317, doi:10.1007/BF03015067, JFM 38.0402.01, S2CID 121688042 (in Italian).
• Wells, Jr., RO (1986), Differential Analysis on Complex Manifolds, Springer-Verlag, ISBN 978-0-387-90419-1