Existence of a Solution to a Vector-valued Ginzburg-landau Equation with a Three Well Potential

where W : R2 → R is positive function with three local minima. A similar result was proved by P.Sternberg in [17] in the case that W has two minima. Moreover, Bronsard, Gui and Schatzman ([5]) proved existence of solution to (1)-(2) when W is equivariant by the symmetry group of the equilateral triangle. Our interest in this problem is originated in some models of three-boundary motion. Material scientists working on transition have found that the motion of grain boundaries is governed by its local mean curvature (see [12],[13] for example). These models naturally arise as the singular limit of the parabolic Ginzburg-Landau equation (see [1]). The relation between grain boundaries motion and the parabolic Ginzburg Landau equation can be described as follows: consider a positive potential W : Ω ⊂ R → R with a finite number of minima {ci}i=i. Let uǫ : R → R be a solution to