# Explicit Relaxation of a Two-Well Hadamard Energy

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## Abstract

We compute an explicit quasiconvex envelope for a subclass of double-well Hadamard energies which model materials undergoing isotropic-to-isotropic elastic phase transitions. The construction becomes possible because of stability of the entire jump set, representing points that can coexist at phase boundaries. To prove stability we apply a recently developed technique for establishing polyconvexity of points on the jump set.

## Keywords

Quasiconvexity Polyconvexity Rank-one convexity Nonlinear elasticity Hadamard material Jump set Elastic stability Binodal## Mathematics Subject Classification

74A50 74G65 49K40 49S05## Notes

### Acknowledgements

This work was started during the stay of YG at ESPCI, Paris supported by Chair Joliot. The work of both authors was supported by the National Science Foundation under Grant No. DMS-1714287. LT was supported additionally by the French Government under the Grant No. ANR-10-IDEX-0001-02 PSL.

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