We have studied the problem of determining part of the boundary of a domain where a potential satisfies the Laplace equation. The potential and its normal derivative have prescribed values on the known part of the boundary that encloses while its normal derivative must vanish on the remaining part. We establish a sufficient condition for the potential to be monotonic along the unknown boundary. This allows us to use the potential to parametrize the boundary. Two methods are presented that solve the problem under this assumption. The first one solves the problem in a closed form and it can be used to define a parameter that will describe the ill-posedness of the problem. The effect of this parameter on the second method presented has been determined for a particular numerical example.

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The inverse problem of finding the shape of the pearlite–austenite interface in a pearlite transformation in steel during slow cooling is considered. Under certain assumptions it is possible to find a hodograph transformation that will generate a non-linear differential equation for the determination of the boundary. For a small interface speed, such an equation can be linearized and solved exactly by means of Fourier series. Some numerical examples as well as asymptotic methods at critical points in the hodograph plane are included.