View source for Partial differential equations, property C for

Property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300302.png" /> stands for  "completeness"  of the set of products of solutions to homogeneous linear partial differential equations. It was introduced in [[#References|[a1]]] and used in [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a13]]] as a powerful tool for proving uniqueness results for many multi-dimensional inverse problems, in particular, inverse scattering problems (cf. also [[Inverse scattering, multi-dimensional case|Inverse scattering, multi-dimensional case]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300303.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300305.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300307.png" /> is a multi-index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300308.png" />, derivatives being understood in the distributional sense, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003010.png" />, are certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003011.png" /> functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003012.png" /> is the null-space of the formal [[Differential operator|differential operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003013.png" />, and the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003014.png" /> is understood in the distributional sense.

Consider the subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003016.png" /> for which the products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003017.png" /> are defined, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003019.png" />.

The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003020.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003022.png" /> if and only if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003023.png" /> is total (complete) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003024.png" />, (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003025.png" /> is fixed), that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003026.png" /> and

<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/p/p130/p130030/p13003027.png" /></td> </tr></table>

<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/p/p130/p130030/p13003028.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003029.png" />.

By property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003031.png" /> one often means property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003032.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003033.png" /> with any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003034.png" />.

Is property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003035.png" /> generic for a pair of formal partial differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003037.png" />?

For the operators with constant coefficients, a necessary and sufficient condition is given in [[#References|[a10]]] for a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003038.png" /> to have property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003039.png" />. For such operators it turns out that property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003040.png" /> is generic and holds or fails to hold simultaneously for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003041.png" />: Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003042.png" />. Denote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003044.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003046.png" />.

Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003047.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003048.png" />.

Define the algebraic varieties (cf. also [[Algebraic variety|Algebraic variety]])

<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/p/p130/p130030/p13003049.png" /></td> </tr></table>

One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003050.png" /> is transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003051.png" />, and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003052.png" />, if and only if there exist a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003053.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003054.png" /> such that the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003055.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003056.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003057.png" />) at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003058.png" /> and the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003059.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003060.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003061.png" /> are transversal (cf. [[Transversality|Transversality]]).

The following result is proved in [[#References|[a1]]]: The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003062.png" /> of formal partial differential operators with constant coefficients has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003063.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003064.png" />.

Thus, property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003065.png" /> fails to hold for a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003066.png" /> of formal differential operators with constant coefficients if and only if the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003067.png" /> is a union of parallel hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003068.png" />.

Therefore, property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003069.png" /> for partial differential operators with constant coefficients is generic.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003070.png" /> and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003071.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003072.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003073.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003075.png" />.

==Examples.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003077.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003078.png" />. It is easy to check that there are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003080.png" /> at which the tangent hyperplanes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003081.png" /> are not parallel. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003082.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003083.png" />. This means that the set of products of harmonic functions in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003084.png" /> is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003086.png" /> (cf. also [[Harmonic function|Harmonic function]]). Similarly one checks that the operators

<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/p/p130/p130030/p13003087.png" /></td> </tr></table>

have property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003088.png" />.

Numerous applications of property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003089.png" /> to inverse problems can be found in [[#References|[a1]]].

Property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003090.png" /> holds for a pair of Schrödinger operators with potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003092.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003093.png" /> is the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003094.png" /> functions with compact support{} (cf. also [[Schrödinger equation|Schrödinger equation]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003099.png" /> is the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030100.png" />, are the scattering solutions corresponding to the Schrödinger operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030103.png" />, then the set of products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030105.png" /> is fixed, is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030107.png" /> is an arbitrary fixed bounded domain [[#References|[a1]]]. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030109.png" /> is fixed, is total in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030110.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030111.png" /> is the [[Sobolev space|Sobolev space]] [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Ramm,   "Multidimensional inverse scattering problems" , Longman/Wiley  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.G. Ramm,   "Scattering by obstacles" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,   "Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering"  ''Inverse Probl.'' , '''3'''  (1987)  pp. L77–L82</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,   "Multidimensional inverse problems and completeness of the products of solutions to PDE"  ''J. Math. Anal. Appl.'' , '''134''' :  1  (1988)  pp. 211–253  (Also: 139 (1989), 302)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.G. Ramm,   "Recovery of the potential from fixed energy scattering data"  ''Inverse Probl.'' , '''4'''  (1988)  pp. 877–886  (Also: 5 (1989), 255)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.G. Ramm,   "Multidimensional inverse problems: Uniqueness theorems"  ''Appl. Math. Lett.'' , '''1''' :  4  (1988)  pp. 377–380</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.G. Ramm,   "Multidimensional inverse scattering problems and completeness of the products of solutions to homogeneous PDE"  ''Z. Angew. Math. Mech.'' , '''69''' :  4  (1989)  pp. T13–T22</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.G. Ramm,   "Property C and uniqueness theorems for multidimensional inverse spectral problem"  ''Appl. Math. Lett.'' , '''3'''  (1990)  pp. 57–60</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.G. Ramm,   "Completeness of the products of solutions of PDE and inverse problems"  ''Inverse Probl.'' , '''6'''  (1990)  pp. 643–664</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.G. Ramm,   "Necessary and sufficient condition for a PDE to have property C"  ''J. Math. Anal. Appl.'' , '''156'''  (1991)  pp. 505–509</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.G. Ramm,   "Property C and inverse problems" , ''ICM-90 Satellite Conf. Proc. Inverse Problems in Engineering Sci.'' , Springer  (1991)  pp. 139–144</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A.G. Ramm,   "Stability estimates in inverse scattering"  ''Acta Applic. Math.'' , '''28''' :  1  (1992)  pp. 1–42</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  A.G. Ramm,   "Stability of solutions to inverse scattering problems with fixed-energy data"  ''Rend. Sem. Mat. e Fisico''  (2001)  pp. 135–211</TD></TR></table>