Polyconvex double well functions

TL;DR

This paper characterizes when a double well function related to phase transitions is polyconvex based on the singular values of matrix differences, enabling existence and uniqueness results for related minimization problems.

Contribution

It provides a precise singular value condition for polyconvexity of double well functions, linking matrix analysis with variational problem solvability.

Findings

Polyconvexity characterized by singular value conditions.

Decomposition into convex and null Lagrangian parts.

Existence and uniqueness of minimizers proven.

Abstract

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. Polyconvexity of $f$ is related to the singular values of the matrix difference $X\_1 - X\_2$. We prove that $f$ is polyconvex if and only if the square of the largest singular value does not exceed the sum of the squares of the other singular values. This condition allows the function to be decomposed into the sum of a strictly convex part and a null Lagrangean. As a direct application of this result, we prove an existence and uniqueness theorem for the corresponding Dirichlet minimization problem of the integral functional.