Polyconvex double well functions

TL;DR

The paper analyzes the polyconvexity of the double-well energy $f(X)=|X-X_1|^2|X-X_2|^2$ by linking polyconvexity to the singular values of $X_1-X_2$, showing that $f$ is polyconvex iff $\sigma_1^2\le\sum_{i=2}^n\sigma_i^2$ where $\sigma_i$ are the singular values. It then proves a decomposition of $f$ into a strictly convex part plus a null Lagrangian, enabling a well-posed Dirichlet minimization problem for the associated energy. The authors analyze the $2\times2$ and $3\times3$ cases, deriving frame-invariant reformulations and exposing when related variants fail polyconvexity via rank-one convexity arguments. Collectively, these results provide a rigorous criterion for well-posed variational problems with double-well energies in nonlinear elasticity, grounded in spectral data of $X_1-X_2$.

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.

Theorems & Definitions (17)

• Lemma 1
• proof
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Corollary 1: Isotropy
• proof
• Lemma 4