2021/04/03 by Lorent, Andrew, Peng, Guanying · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2104.01467
The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose Γ-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on H1-rectifiable sets in 2D, is arguably the key missing part for a proof of the full Γ-convergence of the Aviles-Giga functional. In the first part of this work, for p∈ (1,(4)/(3)] we establish an Lp version of the main theorem of Ghiraldin and Lamy [Comm. Pure Appl. Math. 73 (2020), no. 2, 317-349]. Specifically we show that if m is a solution to the Eikonal equation, then m∈ B(1)/(3)3p,∞,loc is equivalent to all entropy productions of m being in Lploc. This result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in Lploc are rigid) - the rigidity/flexibility threshold of the Eikonal equation is exactly the space B(1)/(3)3,∞,loc. In the second part of this paper, under the assumption that all entropy productions are in Lploc, we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the Lp setting - this is a strong extension of an earlier result of the authors [Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 35 (2018), no. 2, 481-516].