Polyconvex double well functions

We investigate polyconvexity of the double well function $f(X)\,:= |X-X_1|^2|X-X_2|^2$ for given matrices $X_1, X_2 \in R^{n \times n}$. Such functions are fundamental in the modeling of phase transitions in materials, but their non-convex nature presents challenges for the analysis of variational problems. We prove that $f$ is polyconvex if and only if the singular values of the matrix difference $X_1 - X_2$ are all equal. This condition allows the function to be decomposed into the sum of a strictly convex part and a null Lagrangian. As a direct application of this result, we prove an existence and uniqueness theorem for the corresponding Dirichlet minimization problem in the calculus of variations, demonstrating that under this specific condition, there is a unique minimizer for general boundary data.