# 尤拉方程的两个自由边界问题的相容性The Compatibility of Two Free Boundary Problem of Euler Equation

Abstract: In this paper, we model the free boundary problem of the Perpetual American Option as boundary value problem with multiple (or single) singular points in the semi infinite domain, and introduce the generalized characteristic function method to be able to obtain the exact solution of the mathematical model of multiple singular point. In the single singular point case, our solution function takes the maximum value at the singular point. We deduce the consistency condition of the left and right free boundary problem. Under the compatibility condition, the three points, the left and right free boundary points and singular point are the same, so that they all are the optimal implementation point of the Perpetual American Option. In the case of multiple singular points, the conditional judgment of the left and right free boundary points to be the optimal or nearly optimal implementation point is obtained.

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