Factorization for entropy production of the Eikonal equation and regularity

Abstract

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 H 1 -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 L p version of the main theorem of [GL20]. 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 L p loc . Given the main result of [GL20], 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 L p loc 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 L p loc , 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 L p setting -this is a strong extension of the main result of [LP18].

Characterization of minimizers of Aviles-Giga functionals in special domains

“…Among other results, the same expression for σ 0,x has been obtained very recently in [LP21] under the additional assumption that div Φ(m) ∈ L p (Ω) for every Φ ∈ E. As the authors point out, it is still not known if this additional assumption is sufficient to establish that indeed σ 0 vanishes. 1.3.…”

Variational analysis of the $J_1$-$J_2$-$J_3$ model: a non-linear lattice version of the Aviles-Giga functional

“…Already the stronger regularity ∇ϕ ∈ B 1/3 3,q for q < ∞ yields rigidity. In [38], the authors raise the question whether B 1/3 3p,∞ regularity, p > 1 , triggers this rigidity, too. As a partial result, they prove that this regularity implies that the entropy productions div(Φ • ∇ ⊥ ϕ) belong to L p , which they conjecture to be enough to deduce rigidity.…”

“…for all entropies Φ . Moreover, according to [41], a preliminary result on the question whether this can be extended to the case of measures is available as a consequence of recent developments, specifically, the eikonal equation's kinetic formulation established in [28], a Lagrangian representation method [13,39,43,40,42], and ideas used in [38]. The precise statement requires the introduction of a subclass of parametrized entropies {Φ f : f : S 1 → R} (cf.…”

Nonlinear approximation of 3D smectic liquid crystals: sharp lower bound and compactness

“…The compactness and form of the limiting energy, however, are still open. The Aviles-Giga model and related topics such as the eikonal equation and other line-energy models have continued to be active areas of research in the past two decades; see [1,14,20,24,25,26,27,35,36,37,38,39,40,41,50,51] and the references therein.…”