Work

Polyconvex double well functions

Year: 2026

Type: article

Abstract: We investigate polyconvexity of the double well function $$f(X):= |X-X_1|^2|X-X_2|^2$$ f ( X ) : = | X - X 1 | 2 | X - X 2 | 2 for given matrices $$X_1, X_2 \in \mathbb {R}^{n \times n}$$ X 1 , X 2 ∈ R n × 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 $$X_1 - X_2$$ 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.

Source: Calculus of Variations and Partial Differential Equations

DOI: 10.1007/s00526-026-03262-z

Authors: Didier Henrion, Martin Kružík