View source for Partial differential equations, property C for

Property $C$ 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 $D$ be a bounded domain in $\textbf{R}^n$, $n\geq 2$, let $L_mu(x):=\sum^{J}_{|j|=0}a_{jm}(x)D^ju(x)$, where $j$ is a multi-index, $D^ju=\partial^{|j|}u/\partial x_1^{j_{1}}...\partial x_n^{j_{n}}$, derivatives being understood in the distributional sense, the $a_{jm}(x)$, $m=1,2$, are certain $L^{\infty}(D)$ functions, $N_m:=\{w:L_mw=0\text{in}D\}$ is the null-space of the formal [[Differential operator|differential operator]] $L_m$, and the equation $L_mw=0$ is understood in the distributional sense.

Consider the subsets $\widetilde{N}_1\in N_2$ and $\widetilde{N}_2\in N_n$ for which the products $w_1w_2$ are defined, $w_1\in\widetilde{N}_1$, $w_2\in\widetilde{N}_2$.

The pair $\{L_1,L_2\}$ has property $C_p$ if and only if the set $\{w_1w_2\}_{\forall w_{m}\in\widetilde{N}_{m}}$ is total (complete) in $L^p(D)$, ($p\geq 1$ is fixed), that is, if $f(x)\in L^p(D)$ and

\begin{equation}\int_{D}f(x)w_1(x)w_2(x)dx=0,\\\forall w_1\in\widetilde{N}_1,\forall w_2\in\widetilde{N}_2,\end{equation}

then $f(x)\equiv 0$.

By property $C$ one often means property $C_2$ or $C_p$ with any fixed $p\geq1$.

Is property $C$ generic for a pair of formal partial differential operators $L_1$ and $L_2$?

For the operators with constant coefficients, a necessary and sufficient condition is given in [[#References|[a10]]] for a pair $\{L_1,L_2\}$ to have property $C$. For such operators it turns out that property $C$ is generic and holds or fails to hold simultaneously for all $p\in[1,\infty)$: Assume $a_{jm}(x)=a_{jm}=\text{const}$. Denote $L_m(z):=\sum^{J}_{|j|=0}a_{jm}z^j$, $z\in\textbf{C}^n$. Note that $L_m(e^{zx})=e^{zx}L_m(z)$, $z.x:=\sum^n_{j=1}z_jx_j$.

Therefore $e^{zx}\in\widetilde{N}_m$ if and only if $L_m(z)=0$.

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

\begin{equation}\mathcal{L}_m:=\{z:z\in\mathbf{C}^n,L_m(z)=0\}.\end{equation}

One says that $\mathcal{L}_1$ is transversal to $\mathcal{L}_2$, and writes $\mathcal{L}_1\nparallel\mathcal{L}_2$, if and only if there exist a point $\zeta\in\mathcal{L}_1$ and a point $\xi\in\mathcal{L}_2$ such that the tangent space $T_1$ to $\mathcal{L}_1$ (in $\mathbf{C}^n)$ at the point $\zeta$ and the tangent space $T_2$ to $\mathcal{L}_2$ at the point $\xi$ are transversal (cf. [[Transversality|Transversality]]).

The following result is proved in [[#References|[a1]]]: The pair $\{L_1,L_2\}$ of formal partial differential operators with constant coefficients has property $C$ if and only if $\mathcal{L}_1\nparallel\mathcal{L}_2$.

Thus, property $C$ fails to hold for a pair $\{L_1,L_2\}$ of formal differential operators with constant coefficients if and only if the variety $\mathcal{L}_1\cup\mathcal{L}_2$ is a union of parallel hyperplanes in $\mathbf{C}^n$.

Therefore, property $C$ for partial differential operators with constant coefficients is generic.

If $L_1=L_2=L$ and the pair $\{L,L\}$ has property $C$, then one says that $L$ has property $C$.

==Examples.==
Let $n\geq 2$, $L=\nabla^2:=\sum^n_{j=1}$. Then $. 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>