Difference between revisions of "Partial differential equations, property C for"
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Is property $C$ generic for a pair of formal partial differential operators $L_1$ and $L_2$? | 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 | + | 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 | + | 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]]) | 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 | + | 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 | + | 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 | + | 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 | + | Therefore, property $C$ for partial differential operators with constant coefficients is generic. |
− | If | + | If $L_1=L_2=L$ and the pair $\{L,L\}$ has property $C$, then one says that $L$ has property $C$. |
==Examples.== | ==Examples.== | ||
− | Let | + | 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> | <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> |
Revision as of 15:37, 26 April 2021
Property $C$ stands for "completeness" of the set of products of solutions to homogeneous linear partial differential equations. It was introduced in [a1] and used in [a2], [a3], [a4], [a5], [a6], [a7], [a8], [a9], [a10], [a11], [a12], [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).
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 $L_m$, 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 [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)
\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).
The following result is proved in [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 and
at which the tangent hyperplanes to
are not parallel. Thus
has property
. This means that the set of products of harmonic functions in a bounded domain
is complete in
,
(cf. also Harmonic function). Similarly one checks that the operators
![]() |
have property .
Numerous applications of property to inverse problems can be found in [a1].
Property holds for a pair of Schrödinger operators with potentials
,
, where
is the set of
functions with compact support{} (cf. also Schrödinger equation).
If ,
,
,
,
is the unit sphere in
, are the scattering solutions corresponding to the Schrödinger operators
,
,
, then the set of products
,
is fixed, is complete in
, where
is an arbitrary fixed bounded domain [a1]. The set
, where
is fixed, is total in the set
, where
is the Sobolev space [a1].
References
[a1] | A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992) |
[a2] | A.G. Ramm, "Scattering by obstacles" , Reidel (1986) |
[a3] | A.G. Ramm, "Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering" Inverse Probl. , 3 (1987) pp. L77–L82 |
[a4] | 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) |
[a5] | A.G. Ramm, "Recovery of the potential from fixed energy scattering data" Inverse Probl. , 4 (1988) pp. 877–886 (Also: 5 (1989), 255) |
[a6] | A.G. Ramm, "Multidimensional inverse problems: Uniqueness theorems" Appl. Math. Lett. , 1 : 4 (1988) pp. 377–380 |
[a7] | 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 |
[a8] | A.G. Ramm, "Property C and uniqueness theorems for multidimensional inverse spectral problem" Appl. Math. Lett. , 3 (1990) pp. 57–60 |
[a9] | A.G. Ramm, "Completeness of the products of solutions of PDE and inverse problems" Inverse Probl. , 6 (1990) pp. 643–664 |
[a10] | A.G. Ramm, "Necessary and sufficient condition for a PDE to have property C" J. Math. Anal. Appl. , 156 (1991) pp. 505–509 |
[a11] | A.G. Ramm, "Property C and inverse problems" , ICM-90 Satellite Conf. Proc. Inverse Problems in Engineering Sci. , Springer (1991) pp. 139–144 |
[a12] | A.G. Ramm, "Stability estimates in inverse scattering" Acta Applic. Math. , 28 : 1 (1992) pp. 1–42 |
[a13] | A.G. Ramm, "Stability of solutions to inverse scattering problems with fixed-energy data" Rend. Sem. Mat. e Fisico (2001) pp. 135–211 |
Partial differential equations, property C for. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_differential_equations,_property_C_for&oldid=51703