×

Some equations of non-geometrical optics. (English) Zbl 1131.35337

Lupo, Daniela (ed.) et al., Nonlinear equations: methods, models and applications. Based on the workshop on nonlinear analysis and applications, Bergamo, Italy, July 9–13, 2001. Basel: Birkhäuser (ISBN 3-7643-0398-0/hbk). Prog. Nonlinear Differ. Equ. Appl. 54, 257-267 (2003).
The author presents an overview of his results with Talenti concerning equations arising in generalizations of geometrical optics. He sketches some lineaments of the system \[ u|\nabla u|^2-|\nabla v|^2+n^2=0,\quad \nabla u\cdot\nabla v=0 \] where \(u\) and \(v\) are the unknown scalar real-valued fields while \(n=n(x)\) is the refractive index.
The author presents properties of this system in two space dimensions and discusses regions where the solution \((u,v)\) is either elliptic or hyperbolic by deriving certain second-order semilinear PDEs. For the proofs he refers to [R. Magnanini and G. Talenti, Contemp. Math. 238, 203–229 (1999; Zbl 0940.35100) and SIAM J. Math. Anal. 34, No. 4, 805–835 (2003; Zbl 1126.35367)].
Finally he studies a second-order elliptic-parabolic PDE closely related to that theory [cf. R. Magnanini and G. Talenti, in: Romanov, V. G. (ed.) et al., Ill-posed and inverse problems. Dedicated to Academician Mikhail Mikhailovich Lavrentiev on the occasion of his 70th birthday. Utrecht: VSP, 291–304 (2002; Zbl 1060.35089)].
For the entire collection see [Zbl 1020.00013].

MSC:

35J60 Nonlinear elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A05 Geometric optics
PDFBibTeX XMLCite