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| In [[mathematics]], and specifically the field of [[partial differential equations]] (PDEs), a '''parametrix''' is an approximation to a [[fundamental solution]] of a PDE, and is essentially an approximate inverse to a differential operator.
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| 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.
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| ==Overview and informal definition==
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| It is useful to start reviewing what a fundamental solution for a [[differential operator]] ''P''(''D'') with constant coefficients is: it is a [[distribution (mathematics)|distribution]] ''u'' on ℝ<sup>''n''</sup> such that
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| :<math>P(D){u(x)} = \delta(x),\,</math>
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| in the [[weak derivative|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 | |
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| :<math>P(x,D){u(x)} = \delta(x) + \omega(x),\,</math>
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| where ω is some C<sup>∞</sup> function with compact support. The parametrix is a useful concept in the study of [[elliptic differential operator]]s and, more generally, of [[hypoelliptic]] [[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]] of constant coefficient [[differential operator]]s.</ref> and be a [[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]] 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<ref>See the entry about the [[Partial differential operator#Regularity problem|regularity problem for partial differential operators]].</ref> and other qualitative properties
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| ==Parametrices for pseudodifferential operators==
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| More generally, if ''L'' is any pseudodifferential operator of order ''p'', then another pseudodifferential operator ''L''<sup>+</sup> of order ''–p'' is called a '''parametrix''' for ''L'' if the operators
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| :<math>L\circ L^+ - I,\quad L^+\circ L -I</math>
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| are both pseudodifferential operators of negative order. The operators ''L'' and ''L''<sup>+</sup> will admit continuous extensions to maps between the Sobolev spaces ''H''<sup>''s''</sup> and ''H''<sup>''s''+k</sup>. On a compact manifold, the differences above are [[compact operator]]s. In this case the original operator ''L'' defines a [[Fredholm operator]] between the Sobolev spaces.<ref>{{harvnb|Hörmander|1985}}</ref>
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| ==Hadamard parametrix construction==
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| 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]].
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| In the case of the heat equation or the wave equation, where there is a distinguished time parameter ''t'',
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| Hadamard's method consists in taking the fundamental solution
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| 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.<ref>{{harvnb|Hörmander|1985|pp=30–41}}</ref><ref>{{harvnb|Hadamard|1932}}</ref>
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| ==Construction of a fundamental solution from a parametrix==
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| 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}}.
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| If ''L'' is an element of a ring with multiplication * such that
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| :<math>L*P=1+R</math> | |
| for some approximate right inverse ''P'' and "sufficiently small" remainder term ''R'' then, at least formally,
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| :<math> L*P*(1-R+R*R-R*R*R+\cdots) = 1</math>
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| so if the infinite series makes sense then ''L'' has a right inverse
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| :<math>P-P*R+P*R*R-P*R*R*R+\cdots</math>.
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| 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''.
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| ==Notes==
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| {{reflist|30em}}
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| ==References==
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| *{{springer|id=p/p071570|title=Parametrix method|year=2001|first=A.|last=Bejancu}}
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| *{{Citation
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| | last1=Berger
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| | first1=Marcel
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| | author1-link=Marcel Berger
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| | last2=Gauduchon
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| | first2=Paul
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| | last3=Mazet
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| | first3=Edmond
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| | title=Le spectre d'une variété riemannienne
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| | language= [[French language|French]]
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| | place=Berlin, New York
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| | publisher=[[Springer-Verlag]]
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| | series=Lecture Notes in Mathematics
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| | volume=194
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| | pages=VII, 251
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| | doi=10.1007/BFb0064643
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| | year=1971
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| | mr=0282313
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| | zbl= 0223.53034}}
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| *{{Citation
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| | last1=Hadamard
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| | first1=Jacques
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| | title=Lectures on Cauchy's problem in linear partial differential equations
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| | origyear=1923
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| | url=http://books.google.com/books?id=B25O-x21uqkC
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| | place= New York
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| | publisher=[[Dover Publications]]
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| | series=Dover Phoenix editions
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| | isbn=978-0-486-49549-1
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| | year=2003
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| | jfm=49.0725.04
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| | mr=0051411
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| | zbl=0049.34805}}
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| * {{citation
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| |first=J.
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| |last=Hadamard
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| |authorlink=Jacques Hadamard
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| |title=Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques
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| |language= [[French language|French]]
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| |year=1932
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| |publisher=Herman
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| |place=Paris
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| |jfm=58.0519.16
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| |zbl=0006.20501}}.
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| * {{citation
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| |first=L.
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| |last= Hörmander
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| |authorlink=Lars Hörmander
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| |title=The analysis of linear partial differential operators I
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| |series= Grundlehren der Mathematischen Wissenschaft
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| |volume= 256
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| |place=Heidelberg – Berlin – New York
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| |publisher= [[Springer Verlag]]
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| |year=1983
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| |isbn=3-540-12104-8
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| |mr=0717035
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| |zbl= 0521.35001
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| }}.
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| *{{citation
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| |first=L.
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| |last= Hörmander
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| |authorlink=Lars Hörmander
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| |title=The analysis of linear partial differential operators III
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| |series= Grundlehren der Mathematischen Wissenschaft
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| |volume= 274
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| |place=Heidelberg – Berlin – New York
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| |publisher= [[Springer Verlag]]
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| |year=1985
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| |isbn=3-540-13828-5
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| |mr=MR0781536
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| |zbl= 0601.35001
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| }}.
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| *{{citation
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| |first=Eugenio Elia
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| |last=Levi
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| |author-link= Eugenio Elia Levi
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| |title=Sulle equazioni lineari alle derivate parziali totalmente ellittiche
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| |journal=Rendiconti della Reale Accademia dei Lincei, Classe di Scienze Fisiche, Matematiche, Naturali
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| |series= Serie V,
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| |volume=16
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| |issue=12
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| |year=1907
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| |pages=932–938
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| |jfm=38.0403.01
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| }} (in [[Italian language|Italian]]).
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| *{{citation
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| |first=Eugenio Elia
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| |last=Levi
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| |author-link= Eugenio Elia Levi
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| |title=Sulle equazioni lineari totalmente ellittiche alle derivate parziali
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| |journal=[[Rendiconti del Circolo Matematico di Palermo]]
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| |volume=24
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| |issue= 1
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| |year=1907
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| |pages=275–317
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| |doi=10.1007/BF03015067
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| |jfm=38.0402.01
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| }} (in [[Italian language|Italian]]).
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| * {{citation|first=RO|last=Wells, Jr.|title=Differential Analysis on Complex Manifolds|publisher=Springer-Verlag|year=1986|isbn=978-0-387-90419-1}}
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| [[Category:Fourier analysis]]
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| [[Category:Partial differential equations]]
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