View source for Malgrange-Ehrenpreis theorem

Briefly said: Every constant-coefficients [[Linear partial differential equation|linear partial differential equation]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200901.png" /> can be solved.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200902.png" /> be a [[Polynomial|polynomial]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200903.png" /> variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200904.png" />, where the sum is a finite one, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200905.png" /> is a multi-index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200906.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200907.png" />. To <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200908.png" /> one associates a constant-coefficient linear partial differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m1200909.png" />, obtained by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009010.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009012.png" />).

Although several special cases had been known for a long time, it is only in the mid 1950s that it has been shown that for every non-zero constant-coefficient linear partial differential operator the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009013.png" /> can be solved for all functions or distributions (cf. also [[Generalized function|Generalized function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009015.png" />. In particular, there is a distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009018.png" /> is the Dirac mass at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009019.png" /> (cf. also [[Dirac delta-function|Dirac delta-function]]). Such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009020.png" /> is called a fundamental solution. This is the celebrated Malgrange–Ehrenpreis theorem (established independently by B. Malgrange [[#References|[a8]]] and L. Ehrenpreis [[#References|[a1]]]). In case of the [[Laplace operator|Laplace operator]], the [[Newton potential|Newton potential]] (up to some factor) is a fundamental solution.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009021.png" /> is any distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009022.png" /> with compact support, by using a fundamental solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009023.png" /> it is immediate to solve the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009024.png" />. Indeed (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009025.png" /> denoting convolution, cf. also [[Convolution of functions|Convolution of functions]]),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009026.png" /></td> </tr></table>

The Malgrange–Ehrenpreis theory goes further. One can solve the equation for non-compactly supported right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009027.png" /> by exhausting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009028.png" /> by balls, and by solving the equation on each ball. The process is made to converge by corrections based on the approximation of local solutions to the homogeneous equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009029.png" /> by global solutions. The space of solutions to a linear homogeneous ordinary differential equation, with constant coefficients (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009030.png" />), is spanned by exponential solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009031.png" /> (in the example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009032.png" /> is a root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009033.png" />), or by exponential and possibly polynomial-exponential solutions (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009034.png" /> in the case of a double root). Similarly, the solutions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009035.png" /> can be approximated, on convex sets, by linear combinations of polynomial-exponential solutions, functions of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009037.png" /> is a polynomial and necessarily <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009038.png" />. See [[#References|[a2]]], [[#References|[a8]]], [[#References|[a5]]], (7.3.6). More generally, convolution equations can be considered.

Most often, the Malgrange–Ehrenpreis theorem is attacked by using the [[Fourier transform|Fourier transform]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009039.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009040.png" />-function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009041.png" />, say with compact support, its Fourier transform is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009042.png" /></td> </tr></table>

and one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009043.png" />. Looking for a fundamental solution basically consists in solving the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009044.png" />. This is of course totally elementary in case the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009045.png" /> is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009046.png" />, since then, by inverse Fourier transformation, one can define (classically) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009047.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009048.png" /></td> </tr></table>

(For example, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009049.png" /> denoting the usual Laplacian, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009050.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009051.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009052.png" />.)

In general, one has to face two problems: the lack of integrability at infinity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009053.png" />, and the real zeros of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009054.png" />. Roughly speaking, since the two problems are not disjoint, one overcomes the first difficulty by the calculus of distributions, and the second one by switching to the complex domain. Both Malgrange and Ehrenpreis used the [[Hahn–Banach theorem|Hahn–Banach theorem]]. By now (1998) things have been much simplified, and a quick  "constructive"  proof of the Malgrange–Ehrenpreis theorem can be found in [[#References|[a5]]], (7.3.10), see also [[#References|[a3]]], (1.56). A formula claimed to be more explicit was recently given by H. König [[#References|[a7]]].

The solvability of constant-coefficients linear partial differential operators can also be studied with pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009055.png" /> methods. There is a remarkable fundamental result by L.V. Hörmander [[#References|[a4]]], (2.6), whose proof is totally elementary, brief, and a masterpiece:

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009056.png" /> is a bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009057.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009058.png" /> is non-zero, there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009059.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009060.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009061.png" /> with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009062.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009063.png" /></td> </tr></table>

It then follows, from elementary [[Hilbert space|Hilbert space]] theory, that the adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009064.png" /> (associated to the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009065.png" />) is solvable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120090/m12009066.png" />. From there, one can recover the Malgrange–Ehrenpreis theorem. See [[#References|[a10]]], or [[#References|[a9]]] for an elementary exposition. However, the problem being not only to show the existence of a fundamental solution, but to get solutions with good properties, the construction in [[#References|[a5]]], Chap. 7, seems close to being optimal, see [[#References|[a6]]], Chap. 10.

The situation is radically different for operators with variable coefficients (see [[Lewy operator and Mizohata operator|Lewy operator and Mizohata operator]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Ehrenpreis,   "Solutions of some problems of division I"  ''Amer. J. Math.'' , '''76'''  (1954)  pp. 883–903</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Ehrenpreis,   "Solutions of some problems of division II"  ''Amer. J. Math.'' , '''78'''  (1956)  pp. 685–715</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.B. Folland,   "Introduction to partial differential equations" , Princeton Univ. Press  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Hörmander,   "On the theory of general partial differential operators"  ''Acta Math.'' , '''94'''  (1955)  pp. 161–258</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Hörmander,   "The analysis of linear partial differential operators I" , ''Grundl. Math. Wissenschaft.'' , '''256''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Hörmander,   "The analysis of linear partial differential operators II" , ''Grundl. Math. Wissenschaft.'' , '''257''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. König,   "An explicit formula for fundamental solutions of linear partial differential equations with constant coefficients"  ''Proc. Amer. Math. Soc.'' , '''120'''  (1994)  pp. 1315–1318</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B. Malgrange,   "Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution"  ''Ann. Inst. Fourier (Grenoble)'' , '''6'''  (1955/6)  pp. 271–355</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J-P. Rosay,   "A very elementary proof of the Malgrange–Ehrenpreis theorem"  ''Amer. Math. Monthly'' , '''98'''  (1991)  pp. 518–523</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  F. Treves,   "Thèse d'Hörmander"  ''Sém. Bourbaki'' , '''Exp. 130'''  (1956)</TD></TR></table>