Multicritical point

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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.

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 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''n'' such that

:<math>P(D){u(x)} = \delta(x),\,</math>

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

:<math>P(x,D){u(x)} = \delta(x) + \omega(x),\,</math>

where ω is some C∞ 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

==Parametrices for pseudodifferential operators==
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
:<math>L\circ L^+ - I,\quad L^+\circ L -I</math>
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 operator]]s. In this case the original operator ''L'' defines a [[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 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.<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 ''L'' is an element of a ring with multiplication * such that
:<math>L*P=1+R</math>
for some approximate right inverse ''P'' and "sufficiently small" remainder term ''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 ''L'' has a right inverse
:<math>P-P*R+P*R*R-P*R*R*R+\cdots</math>.
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''.

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

==References==
*{{springer|id=p/p071570|title=Parametrix method|year=2001|first=A.|last=Bejancu}}
*{{Citation
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* {{citation
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}}.
*{{citation
|first=L.
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|title=The analysis of linear partial differential operators III
|series= Grundlehren der Mathematischen Wissenschaft
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|place=Heidelberg – Berlin – New York
|publisher= [[Springer Verlag]]
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*{{citation
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|author-link= Eugenio Elia Levi
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|journal=Rendiconti della Reale Accademia dei Lincei, Classe di Scienze Fisiche, Matematiche, Naturali
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|pages=932–938
|jfm=38.0403.01
}} (in [[Italian language|Italian]]).
*{{citation
|first=Eugenio Elia
|last=Levi
|author-link= Eugenio Elia Levi
|title=Sulle equazioni lineari totalmente ellittiche alle derivate parziali
|journal=[[Rendiconti del Circolo Matematico di Palermo]]
|volume=24
|issue= 1
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|doi=10.1007/BF03015067
|jfm=38.0402.01
}} (in [[Italian language|Italian]]).
* {{citation|first=RO|last=Wells, Jr.|title=Differential Analysis on Complex Manifolds|publisher=Springer-Verlag|year=1986|isbn=978-0-387-90419-1}}

[[Category:Fourier analysis]]
[[Category:Partial differential equations]]

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Multicritical point