# Parametrix

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{{Short description|Concept in the solution of linear partial differential equations}}
In [mathematics](/source/mathematics), and specifically the field of [partial differential equations](/source/partial_differential_equations) (PDEs), a '''parametrix''' is an approximation to a [fundamental solution](/source/fundamental_solution) of a PDE, and is essentially an approximate inverse to a differential operator.

A parametrix for a differential operator is often easier to construct than a fundamental solution, and for many purposes is almost as good. It is sometimes possible to construct a fundamental solution from a parametrix by iteratively improving it.

==Overview and informal definition==
It is useful to review what a fundamental solution for a [differential operator](/source/differential_operator) {{math|''P''(''D'')}} with constant coefficients is: it is a [distribution](/source/distribution_(mathematics))  {{math|''u''}} on <math>\mathbb{R}^{n}</math> such that
:<math>P(D){u(x)} = \delta(x)~,</math>
in the [weak sense](/source/weak_derivative), where  {{math|''δ''}} is the [Dirac delta distribution](/source/Dirac_delta_distribution).

In a similar way, a '''parametrix''' for a variable coefficient differential operator  {{math|''P''(''x,D'')}} is a distribution  {{math|''u''}} such that
:<math>P(x,D){u(x)} = \delta(x) + \omega(x) ~,</math>
where  {{math|''ω''}}  is some  {{math|''C'' <sup>∞</sup>}} function with compact support.

The parametrix is a useful concept in the study of [elliptic differential operator](/source/elliptic_differential_operator)s and, more generally, of [hypoelliptic](/source/hypoelliptic) [pseudodifferential operator](/source/pseudodifferential_operator)s with variable coefficient, since for such operators over appropriate domains a parametrix can be shown to exist, can be somewhat easily constructed<ref>By using known facts about the [fundamental solution](/source/fundamental_solution) of constant coefficient [differential operator](/source/differential_operator)s.</ref> and be a [smooth function](/source/smooth_function) away from the origin.<ref>{{harvnb|Hörmander|1983|p=170}}</ref>

Having found the analytic expression of the parametrix, it is possible to compute the solution of the associated fairly general [elliptic partial differential equation](/source/elliptic_partial_differential_equation) by solving an associated [Fredholm integral equation](/source/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<ref>See the entry about the [regularity problem for partial differential operators](/source/Partial_differential_operator).</ref> and other qualitative properties.

==Parametrices for pseudodifferential operators==
More generally, if  {{math|''L''}} is any pseudodifferential operator of order  {{math|''p''}}, then another pseudodifferential operator  {{math|''L''<sup>+</sup>}} of order  {{math|''–p''}} is called a '''parametrix''' for  {{math|''L''}} if the operators
:<math>L\circ L^+ - I,\quad L^+\circ L -I</math>
are both pseudodifferential operators of negative order. The operators  {{math|''L''}} and  {{math|''L''<sup>+</sup>}}  will admit continuous extensions to maps between the Sobolev spaces  {{math|''H''<sup>''s''</sup>}} and  {{math|''H''<sup>''s''+''k''</sup>}}.

On a compact manifold, the differences above are [compact operator](/source/compact_operator)s. In this case the original operator  {{math|''L''}} defines a [Fredholm operator](/source/Fredholm_operator) between the Sobolev spaces.<ref>{{harvnb|Hörmander|1985}}</ref>

==Hadamard parametrix construction==
An explicit construction of a parametrix for second order partial differential operators based on [power series](/source/power_series) developments was discovered by [Jacques Hadamard](/source/Jacques_Hadamard). It can be applied to the  [Laplace operator](/source/Laplace_operator), the [wave equation](/source/wave_equation) and the [heat equation](/source/heat_equation).

In the case of the heat equation or the wave equation, where there is a distinguished time parameter  {{math|''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](/source/formal_power_series) in  {{math|''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](/source/asymptotic_expansion) of the exact solution. A suitable truncation of the power series then yields a parametrix.<ref>{{harvnb|Hörmander|1985|pp=30–41}}</ref><ref>{{harvnb|Hadamard|1932}}</ref>

==Construction of a fundamental solution from a parametrix==

A sufficiently good parametrix can often be used to construct an exact fundamental solution by a convergent iterative procedure as follows {{harv|Berger|Gauduchon|Mazet|1971}}.

If  {{math|''L''}} is an element of a ring with multiplication * such that 
:<math>L*P=1+R</math>
for some approximate right inverse  {{math|''P''}} and "sufficiently small" remainder term  {{math|''R''}} then, at least formally, 
:<math> L*P*(1-R+R*R-R*R*R+\cdots) = 1</math>
so if the infinite series makes sense then  {{math|''L''}} has a right inverse
:<math>P-P*R+P*R*R-P*R*R*R+\cdots</math>.

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

If   {{math|''P''}} and   {{math|''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   
{{math|''L''}} involve  convolution of   {{math|''P''}} with copies of   {{math|''R''}}.

==Notes==
{{reflist|30em}}

==References==
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|language= French
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* {{citation
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*{{citation
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}} (in [Italian](/source/Italian_language)).
*{{citation
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  |journal=[Rendiconti del Circolo Matematico di Palermo](/source/Rendiconti_del_Circolo_Matematico_di_Palermo)
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Category:Fourier analysis
Category:Partial differential equations
Category:Schwartz distributions

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