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Multidimensional inverse problems and completeness of the products of solutions to PDE. (English) Zbl 0632.35076
A 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.
Reviewer: A. G. Ramm

35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
Full Text: DOI
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