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The Eikonal equation arises naturally in the limit of the second orderAviles-Giga functional whose $\Gamma$-convergence is a long standingchallenging problem. The theory of entropy solutions of the Eikonal equationplays a central role in the variational analysis of this problem. Establishingfine structures of entropy solutions of the Eikonal equation, e.g.concentration of entropy measures on $\mathcal{H}^1$-rectifiable sets in $2$D,is arguably the key missing part for a proof of the full $\Gamma$-convergenceof the Aviles-Giga functional. In the first part of this work, for $p\in\left(1,\frac{4}{3}\right]$ we establish an $L^p$ version of the main theoremof 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$m\in B^{\frac{1}{3}}_{3p,\infty,loc}$ is equivalent to all entropy productionsof $m$ being in $L^p_{loc}$. This result also shows that as a consequence of aweak form of the Aviles-Giga conjecture (namely the conjecture that allsolutions to the Eikonal equation whose entropy productions are in $L^p_{loc}$are rigid) - the rigidity/flexibility threshold of the Eikonal equation isexactly the space $ B^{\frac{1}{3}}_{3,\infty,loc}$. In the second part of thispaper, under the assumption that all entropy productions are in $L^p_{loc}$, weestablish a factorization formula for entropy productions of solutions of theEikonal equation in terms of the two Jin-Kohn entropies. A consequence of thisformula is control of all entropy productions by the Jin-Kohn entropies in the$L^p$ setting - this is a strong extension of an earlier result of the authors[Annales de l'Institut Henri Poincar\'{e}. Analyse Non Lin\'{e}aire 35 (2018),no. 2, 481-516].

Factorization for entropy production of the Eikonal equation and regularity