Rigidity of the Eikonal equation subject to two L^p entropy measures
The Aviles-Giga functional arose as a model in connection with the theory of smectic liquid crystals and thin film blisters. It is a second order generalization of the classical Modica-Mortola functional, and constitutes one of the most important open problems in the field of Gamma-convergence. The central tool for the analysis of the problem involves entropy measures. At the heart of the problem, a lack of understanding of the fine structures of entropy measures is a major obstruction towards a full proof of the Gamma-limit conjecture. In a recent work, we make progress towards understanding the absolutely continuous part of entropy measures. This complements existing results to give a deeper understanding of structure of entropy measures. This is joint work with X. Lamy and A. Lorent.