Multidimensional inverse problems and completeness of the products of solutions to PDE.

*(English)*Zbl 0632.35076A method is given to prove uniqueness theorems for some multidimensional inverse problems with data given at a fixed frequency. A number of new uniqueness theorems are proved. In particular:

1) if \(A(\theta ',\theta)\) is given for all \(\theta ',\theta \in S^ 2\) at a fixed \(k>0\) then \(q(x)\) is uniquely determined. Here A(\(\theta\) ’,\(\theta)\) is the scattering amplitude for the equation \(\ell _ qu:=\Delta u+k^ 2u-q(x)u=0\) in \(\mathbb R^ 3\), \(k=\text{const}>0\), \(q(x)=\overline{q(x)}\in L^ 2(D)\), \(D\) is a finite region, \(q=0\) outside \(D\);

2) if \(u(x,y)\) is given for all \(x,y\in P:=\{x: x_ 3=0\}\) at a fixed \(k>0\) then \(v(x)\) is uniquely determined. Here \(L_ vu:=(\Delta +k^ 2+k^ 2v(x))u=-\delta (x-y)\) in \(\mathbb R^ 3\), \(v=\bar v\in L^ 2(D)\), \(D\subset \mathbb R^ 3_ -=\{x: x_ 3<0\}\) is a finite region, \(v=0\) outside \(D\);

3) if \(u(x,y,k)\) is given for all \(x,y\in P\) and all \(k\in (0,\infty)\) then \(v(x)\) and \(h(k)\) are uniquely determined. Here \(L_ vu=-\delta (x-y)h(k)\) and \(h(k)=\int ^{T}_{0}a(t)\exp (ikt)\,dt\), \(a=\bar a\in L^ 2[0,T]\) is the unknown wavelet shape;

4) if the set \(\{u,\sigma u_ N\}\;\forall u\in H^{3/2}(\Gamma)\) is known then \(\sigma(x)\) is uniquely determined in \(D\). Here \(D\subset \mathbb R^ 3\) is a finite region with a smooth boundary \(\Gamma\), \(0<m\leq \sigma (x)\in H^ 2(D)\), \(H^ m\) is the Sobolev space, \(u_ N\) is the normal derivative of \(u\) on \(\Gamma\).

Other theorems are also proved.

The method is based on property C introduced by the author. A pair \((\ell _ 1,\ell _ 2)\), where \(\ell _ j\) are linear PDO has property C iff \(\int _{D}fuw \,dx=0\) \(\forall u\in N_ D(\ell _ 1)\), \(\forall w\in N_ D(\ell _ 2)\) implies \(f=0\). Here \(D\subset \mathbb R^ n\) is a bounded region, \(f\in L^ p(D)\), \(p\geq 1\), \(N_ D(\ell):=\{u: \ell u=0\;\text{in}\;D\}\). An operator \(\ell\) has property C if the pair (\(\ell,\ell)\) has property C. A necessary and sufficient condition for a PDO with constant coefficients to have property C is given. It is proved that \(\ell _ q\) and \((\ell _{q_ 1},\ell _{q_ 2})\) have property C. It is known that property C implies uniqueness. A number of other results and ideas are introduced. It is shown that the set of scattering solutions is complete in \(L^ 2(D)\) in \(N_ D(\ell _ q)\). This result plays an important role in the theory. Numerical methods for reconstruction of \(q(x), v(x)\) and \(\sigma(x)\) from the data are outlined.

1) if \(A(\theta ',\theta)\) is given for all \(\theta ',\theta \in S^ 2\) at a fixed \(k>0\) then \(q(x)\) is uniquely determined. Here A(\(\theta\) ’,\(\theta)\) is the scattering amplitude for the equation \(\ell _ qu:=\Delta u+k^ 2u-q(x)u=0\) in \(\mathbb R^ 3\), \(k=\text{const}>0\), \(q(x)=\overline{q(x)}\in L^ 2(D)\), \(D\) is a finite region, \(q=0\) outside \(D\);

2) if \(u(x,y)\) is given for all \(x,y\in P:=\{x: x_ 3=0\}\) at a fixed \(k>0\) then \(v(x)\) is uniquely determined. Here \(L_ vu:=(\Delta +k^ 2+k^ 2v(x))u=-\delta (x-y)\) in \(\mathbb R^ 3\), \(v=\bar v\in L^ 2(D)\), \(D\subset \mathbb R^ 3_ -=\{x: x_ 3<0\}\) is a finite region, \(v=0\) outside \(D\);

3) if \(u(x,y,k)\) is given for all \(x,y\in P\) and all \(k\in (0,\infty)\) then \(v(x)\) and \(h(k)\) are uniquely determined. Here \(L_ vu=-\delta (x-y)h(k)\) and \(h(k)=\int ^{T}_{0}a(t)\exp (ikt)\,dt\), \(a=\bar a\in L^ 2[0,T]\) is the unknown wavelet shape;

4) if the set \(\{u,\sigma u_ N\}\;\forall u\in H^{3/2}(\Gamma)\) is known then \(\sigma(x)\) is uniquely determined in \(D\). Here \(D\subset \mathbb R^ 3\) is a finite region with a smooth boundary \(\Gamma\), \(0<m\leq \sigma (x)\in H^ 2(D)\), \(H^ m\) is the Sobolev space, \(u_ N\) is the normal derivative of \(u\) on \(\Gamma\).

Other theorems are also proved.

The method is based on property C introduced by the author. A pair \((\ell _ 1,\ell _ 2)\), where \(\ell _ j\) are linear PDO has property C iff \(\int _{D}fuw \,dx=0\) \(\forall u\in N_ D(\ell _ 1)\), \(\forall w\in N_ D(\ell _ 2)\) implies \(f=0\). Here \(D\subset \mathbb R^ n\) is a bounded region, \(f\in L^ p(D)\), \(p\geq 1\), \(N_ D(\ell):=\{u: \ell u=0\;\text{in}\;D\}\). An operator \(\ell\) has property C if the pair (\(\ell,\ell)\) has property C. A necessary and sufficient condition for a PDO with constant coefficients to have property C is given. It is proved that \(\ell _ q\) and \((\ell _{q_ 1},\ell _{q_ 2})\) have property C. It is known that property C implies uniqueness. A number of other results and ideas are introduced. It is shown that the set of scattering solutions is complete in \(L^ 2(D)\) in \(N_ D(\ell _ q)\). This result plays an important role in the theory. Numerical methods for reconstruction of \(q(x), v(x)\) and \(\sigma(x)\) from the data are outlined.

Reviewer: A. G. Ramm

##### Keywords:

uniqueness; multidimensional inverse problems; data given at a fixed frequency; scattering amplitude; constant coefficients; scattering solutions
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\textit{A. G. Ramm}, J. Math. Anal. Appl. 134, No. 1, 211--253 (1988; Zbl 0632.35076)

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##### References:

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