2016/10/02 by Andrew Lorent, Lorent, Andrew, Guanying Peng +1
Engineering · Mathematics · #35F50 #49K21 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1610.00237
openalex publication_date 2016/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Aviles-Giga functional Iε(u)=∫Ω \frac|1-|∇ u|2|2ε+ε|∇2 u|2 dx is a well known second order functional that models phenomena from blistering to liquid crystals. The zero energy states of the Aviles-Giga functional have been characterized by Jabin, Otto, Perthame. Among other results they showed that if limn→ ∞ Iεn(un)=0 for some sequence un∈ W2,20(Ω) and u=limn→ ∞ un then ∇ u is Lipschitz continuous outside a locally finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation |∇ u|=1 a.e. and if for every "entropy" Φ function u satisfies ∇⋅[Φ(∇ u⊥)]=0 distributionally in Ω then ∇ u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result by showing that if Ω is bounded and simply connected, u satisfies the Eikonal equation and if ∇⋅(Σe1 e2(∇ u⊥))=0and∇⋅(Σε1 ε2(∇ u⊥))=0distributionally inΩ, where Σe1 e2 and Σε1 ε2 are the entropies introduced by Ambrosio, DeLellis, Mantegazza, Jin, Kohn, then ∇ u is locally Lipschitz continuous outside a locally finite set.