2021/06/09 by Michael Novack, Xiaodong Yan, Novack, Michael +1
Engineering · Materials Science · Mathematics · #Advanced Materials and Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Liquid Crystal Research Advancements #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2106.05195
openalex publication_date 2021/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the 3D smectic energy Eε( u) =(1)/(2)∫Ω(1)/(ε ) ( uz-\frac( ux)2+( uy)22) 2+ε ( uxx+uyy)2 dx dy dz. The model contains as a special case the well-known 2D Aviles-Giga model. We prove a sharp lower bound on Eε as ε → 0 by introducing 3D analogues of the Jin-Kohn entropies. The sharp bound corresponds to an equipartition of energy between the bending and compression strains and was previously demonstrated in the physics literature only when the approximate Gaussian curvature of each smectic layer vanishes. Also, for εn→ 0 and an energy-bounded sequence \un \ with ‖∇ un‖Lp(Ω), ‖∇ un‖L2(∂ Ω)≤ C for some p>6, we obtain compactness of ∇ un in L2 assuming that Δxyun has constant sign for each n.