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A special type of multi-variate polynomial of degree 4, called the doublewell potential function, is studied. When the function is bounded from below,it has a very unique property that two or more local minimum solutions areseparated by one local maximum solution, or one saddle point. Our intension inthis paper is to categorize all possible configurations of the double wellpotential functions mathematically. In part I, we begin the study with derivingthe double well potential function from a numerical estimation of thegeneralized Ginzburg-Landau functional. Then, we solve the global minimumsolution from the dual side by introducing a geometrically nonlinear measurewhich is a type of Cauchy-Green strain. We show that the dual of the dualproblem is a linearly constrained convex minimization problem, which is mappedequivalently to a portion of the original double well problem subject toadditional linear constraints. Numerical examples are provided to illustratethe important features of the problem and the mapping in between.

Double Well Potential Function and Its Optimization in The n-dimensional Real Space -- Part I