Celebratio Mathematica — Morawetz

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Works connected to Donald A. Ludwig

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C. S. Morawetz and D. Ludwig: “An inequality for the reduced wave operator and the justification of geometrical optics,” Comm. Pure Appl. Math. 21 : 2 (March 1968), pp. 187–203. MR 223136 Zbl 0157.18701 article

We present an inequality for the reduced wave operator in the exterior of a star-shaped surface in \( n \)-space, with a Dirichlet boundary condition on the surface and a radiation condition at infinity. This inequality is used to demonstrate the continuous dependence (in a suitable norm) of the solution of a scattering problem upon the boundary data and inhomogeneous term in the differential equation. This basic result is then used together with the results of D. Ludwig [7] to prove that the formal solution of the scattering problem for a convex body, which is given by geometrical optics, is asymptotic to the exact solution. Similar results have been given in two dimensions by V. S. Buslaev [1] and R. Grimshaw [2], using different methods, who also consider the Neumann problem. Unfortunately the methods used here are inapplicable in that case.

C. S. Morawetz and D. Ludwig: “The generalized Huyghens’ principle for reflecting bodies,” Comm. Pure Appl. Math. 22 : 2 (March 1969), pp. 189–205. MR 605677 Zbl 0167.10102 article