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Viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape-from-shading equations with discontinuous intensities. (English) Zbl 0936.35048
This paper seems to be an extended (and apparently older) version of the communication [D. Ostrov, ISMN, Int. Ser. Numer. Math. 130, 767-772 (1999; Zbl 0932.35040)] in which not only the detailed proofs of the theorems are given but also the physical background and the derivation of the SAR SFS (Synthetic Aperture Radar Shape-From-Shading) equation: \[ u_r -I(y,r)\sqrt{.5+\sqrt{.25+(1+u_y^2)/I(y,r)^2}}=0, \quad u(y,R_1)\equiv u_0(y), \quad y\in \mathbb{R}, \;r\in (R_1,R_2),\tag{1} \] and some examples and numerical results are presented.
In the case the “intensity function” \(I\) is discontinuous the author uses the semicontinuous functions: \(\overline I\), \(\underline I\) and the Lipschitzian “approximants” \(I^\varepsilon\), \(I_\varepsilon\), \(\varepsilon >0\) defined by: \[ \begin{alignedat}{2} \overline I(x)&:=\limsup_{\xi \to x} I(\xi),\qquad& \underline I(x)&:=\liminf_{\xi \to x} I(\xi), \\ I^\varepsilon (x)&:=\sup_{\xi}[\overline I(\xi)-|x-\xi|/\varepsilon],\qquad& I_\varepsilon (x)&:=\inf_{\xi}[\underline I(\xi)+|x-\xi|/\varepsilon],\end{alignedat} \] which, when introduced in (1) produce unique viscosity (Lipschitzian) solutions \(u^\varepsilon\), \(u_\varepsilon\), respectively, which are proved to be monotonic as \( \varepsilon \to 0+\) and therefore convergent to \(\overline u\), \(\underline u,\) taken as upper and, respectively, lower (generalized) solutions. The author gives an example showing that his “generalized solution” \(u= \overline u= \underline u\) may not coincide with Ishii’s viscosity solution of equations defined by discontinuous Hamiltonians in [H. Ishii, Bull. Fac. Sci. Engrg., Chuo Univ. 28, 33-77 (1985)].

MSC:
35F20 Nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35R05 PDEs with low regular coefficients and/or low regular data
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35D05 Existence of generalized solutions of PDE (MSC2000)
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