Brief Related and Historical Comments

The reader may wonder if the SQM limits the class of possible solutions of the FFR problem to solutions that can be represented in terms of supporting paraboloids and limits of sequences of such solutions, similar to the traditional approach limiting the admissible solution to optics with a priori assumed symmetries. This is an important question which does not have a simple answer. Certainly, the surfaces constructed with the SQM are not required to have any symmetries. However, because of the nonlinearities in the partial differential equations describing the problem it is natural to expect several classes of solutions (Oliker, 2008b). In particular, solutions of type В discussed in Section 3.4 form another class of solutions to the FFR problem. Solutions in this class are only piece-wise convex but not globally convex. The difficulties associated with this issue have already been pointed out by B. Kinber (Kinber, 1984). Setting up specific boundary conditions may not always remove the ambiguity and may further complicate the problem. A seemingly reasonable approach is to identify classes of solutions and try to establish uniqueness within such classes. For example, for the FFR problem the class of convex solutions is of particular interest as it may be easier to analyze and fabricate (Malyak, 1992).

Because of important practical applications, the FFR problem has been discussed extensively in engineering literature. Essentially three different approaches have been used to formulate the general problem analytically and solve it numerically. In the first approach the problem is formulated as a system of first order partial differential equations and a method resembling the method of characteristics is applied to solve the system numerically; see Galindo-Israel et al. (1987), Galindo- Israel et al. (1991a), and other references there. In the second approach the same problem is formulated as a boundary value problem for a second order partial differential equation of the Monge-Ampere type for a certain complex-valued function (Westcott, 1983). Then a linearization procedure is used to construct a solution close to some a priori selected solution. In both approaches a rigorous mathematical analysis of the resulting equations was lacking and consequently the validity of the numerics was never fully established. This led to a controversy regarding existence and uniqueness of solutions that has still not been resolved (Kinber, 1984; Galindo-Israel et al., 1987; Galindo-Israel et al., 1991a). The approach presented in previous sections is mathematically rigorous in its formulation, analysis, and

numerics. In the next section this approach is further developed and connected

with calculus of variations.

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