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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

##### MSC:
 35R30 Inverse problems for PDEs 35P25 Scattering theory for PDEs
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