A FOURIER METHOD FOR THE CONSTRUCTION OF CERTAIN EXTREMAL SOLUTIONS OF THE INVERSE DIRICHLET PROBLEM IN 2 DIMENSIONS

The inverse Dirichlet problem for Laplace's equation in two dimensions entails recovery of quantitative information about the boundary value function on a curve C, from a finite collection of samples of a harmo nic function, measured at points in the bounded region S enclosed by C . For all but the simplest curves, a Green function is not readily ava ilable, so the inverse problem is instead cast in terms of the fundame ntal representation theorem relating a harmonic function in S to both its boundary value and outward normal derivative on C. By expressing C as a parametric curve, with a parameter running from 0 to 2 pi as C i s traversed, both of these functions can be expanded as truncated Four ier series, with both sets of coefficients now serving as unknowns in the inverse problem. The representation theorem applied to points on C becomes a set of algebraic relations between the sets of coefficients . Likewise, data values become further constraints among the coefficie nts. Various extremal solutions for the boundary value function can be constructed, optimizing solution functionals which can be expressed a s functions of the coefficients, using Parseval's equality. Lagrange m ultipliers impose the auxiliary constraints relating the coefficient s ets. As examples, several types of extremal solutions are constructed for a single data set, with a bounding curve C easily represented para metrically, but for which the Green function is not easily accessible.