Formulas for the inverse problem for kinetic equation and for integral geometry (Q2724856)

The author is concerned with the problem of recovering the unknown function \(\lambda\) in the nonlinear kinetic equation NEWLINE\[NEWLINE\begin{multlined} D_xw(x,y,p) + pD_yw(x,y,p) + \biggl[ (1+p^2){D_y\mu(x,y)\over \mu(x,y)}-p(1+p^2){D_x\mu(x,y)\over \mu(x,y)}\biggr] D_yw(x,y,p)\\ = f(w(x,y,p))\lambda(x,y)(1+p^2)^{1/2},\qquad (x,y,p)\in {\mathbb R}\times [a,+\infty)\times {\mathbb R},\end{multlined} \tag{1}NEWLINE\]NEWLINE and subject to the initial condition NEWLINE\[NEWLINE w(x,a,p)=w_0(x,p),\qquad (x,y)\in {\mathbb R}^2, NEWLINE\]NEWLINE where \(f:{\mathbb R}\to {\mathbb R}\) and \(\mu:[a,+\infty)\times {\mathbb R}\to {\mathbb R}_+\) are given functions. NEWLINENEWLINENEWLINEWhen \(f\equiv 1\) such a problem is strictly related to the problem in integral geometry of recovering the function \(\lambda\) in the equation NEWLINE\[NEWLINE \int_{\gamma(x,a,p)} \lambda(\xi,\eta)[(d\xi)^2+(d\eta)^2]^{1/2} = F(x,p),\quad (x,p)\in {\mathbb R}\times {\mathbb R}_-, \tag{2}NEWLINE\]NEWLINE \(F\) being a given function. NEWLINENEWLINENEWLINEKnowing two independent integrals of equation (1) with \(f\equiv 0\), the author constructs what we could call a general integral to the identification problem for (1), i.e., a family of pairs \((w,\lambda)\) solving equation (1) and depending on two arbitrary functions in two variables. (Some annoying misprints occur in the basic computation of \(Lw\) on page 11.) NEWLINENEWLINENEWLINE\noindent Using the quoted result the author can recover the function \(\lambda\) in the integral problem (2).