# zbMATH — the first resource for mathematics

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)
Full Text: