這篇文章裡我們提出和研究的一個特殊類型的四次多變量多項式，我們稱之為雙井能量泛函問題。此問題在有下界的情形下，圖形上它將有一個非常獨特的特徵，它可能有兩個或更多個的局部極小值，且被局部極大值或鞍點分離開來。我們從廣義的 Ginzburg-Landau 泛函的數值估計開始研究與推導，並從該問題的對偶問題刻劃出全域最小解。我們證明了原問題的二次對偶問題是一個線性約束凸最小化問題，且二次對偶問題可以等價映射到原始雙井能量問題的部分(或全部)。數值例子提供了本問題的重要特徵與其二次對偶的映射。

A special type of multi-variate polynomial of degree 4, called the double well potential problem, is proposed and studied. It has a very unique feature in the graph that two or more local minima are separated by a local maximum or a saddle point, provided the entire function is bounded from below. We begin the study by deriving the problem from a numerical estimation for the generalized Ginzburg-Landau functional, followed by the characterization for the global minimum solution from the dual. We show that the dual of the dual problem is a linearly constrained convex minimization problem, which is mapped to a portion (maybe the entire) of the original double well potential problem, although the mapping might not be one-to-one. Numerical examples are provided to see the important features of the problem and also the mapping from its dual of the dual.

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