Inverse problems for multivelocity transfer equation in the plane-symmetric case (Q2712503)

The author considers three identification problems related to the following linear integral equation concerning particle transfer through a plane inhomogeneous layer of depth \(H\), where \(X=[0,H]\times [-1,1]\times [0,E_0]\): NEWLINE\[NEWLINE \psi(x,\mu,E) = \int_{X_0} k(x,\mu,E,x',\mu',E')\psi(x',\mu',E') dx' d\mu' dE' + f(x,\mu,E),\quad \forall (x,\mu,E)\in X. \tag{1}NEWLINE\]NEWLINE The kernel \(k\) and the right-hand side \(f\) admit the following representations: NEWLINE\[NEWLINE k(x,\mu,E,x',\mu',E') = {\sigma_s(x',E')\over |\mu|}\exp\Big( - {\tau(x',x,E)\over |\mu|}\Big) g(\mu,E,x',\mu',E')\theta\Big( {x-x'\over \mu}\Big), NEWLINE\]NEWLINE NEWLINE\[NEWLINE f(x,\mu,E) = {1\over |\mu|}\int_0^H \exp\Big( - {\tau(x',x,E)\over |\mu|}\Big) \theta\Big( {x-x'\over \mu}\Big)q(x',\mu,E) dx', NEWLINE\]NEWLINE where \(\tau(x',x,E)=|\int_{x'}^x \sigma(s,E) ds|\) is the optical length, \(\theta\) the Heaviside function, \(\psi\) is the density of the particle flow, \(\sigma\) and \(\sigma_s\) are the nuclear cuts, \(g\) is the dispersion index and \(q\) is the density of sources in the layer. Moreover, functions \(g\), \(q\) and \(\sigma\), \(\sigma_s\) are assumed to belong to \(L^1\) and \(L^\infty\), respectively, \(\sigma\) being a \textit{nowhere} vanishing function. Assume further that, for any fixed \(H_0\in (0,H)\), some of the functions \(\sigma\), \(\sigma_s\), \(g\) and \(q\) are unknown and satisfy the relations NEWLINE\[NEWLINE \sigma_0=q=0\quad \text{ in\;} (H_0,H]\quad \text{ and\;} \sigma(x,E)=\sigma_0(E)\geq m>0,\quad \forall x\in [H_1,H_2]\subset (H_0,H). NEWLINE\]NEWLINE As an additional information the equation NEWLINE\[NEWLINE \int_0^1 \lambda(\mu)\psi(x,\mu,E)d\mu = \ell(x,E),\qquad x\in [H_1,H_2],\;E\in [E_1,E_2]\subset (0,E_0] \tag{2}NEWLINE\]NEWLINE is prescribed, \(\lambda \in L^\infty(0,1)\) being a nowhere vanishing function in \([0,1]\). NEWLINENEWLINENEWLINEThe author proves, under suitable conditions on the data, uniqueness results for the following three identification problems. NEWLINENEWLINENEWLINEProblem 1. Assume that \(\sigma\) is known in \((H_0,H_2]\times [E_1,E_2]\) and \(g\) and \(q\) admit finite Fourier developments with respect to the Legendre polynomials. Determine a pair \((\psi,\tau_0)\) satisfying (1), (2), where \(\tau_0(E)=\int_0^{H_0} \sigma(s,E) ds\), \(E\in [E_1,E_2]\). NEWLINENEWLINENEWLINEProblem 2. Under the assumption that \(g\) and \(q\) are independent of \(\mu\), determine a triplet \((\psi,g,q)\) of summable functions satisfying (1), (2), where the functions \(\sigma(x,E)\), \(\sigma_0(x,E)\) and \(g(x,E,E')\) are known. NEWLINENEWLINENEWLINEProblem 3. Assume that \(\sigma\) is a known function in \(L^\infty((0,H_0]\times [E_1,E_2])\). Determine a pair \((\psi,a)\in L^1(X_0)\times L^\infty([E_1,E_2];L^1([0,H_0]))\) satisfying (1), (2), where \(X_0=[0,H_0]\times [0,1]\times [E_1,E_2]\), \(a=0\) in \((H_0,H]\times [E_1,E_2]\) and \(a\) admits a special representation.