2022/01/25 by Qiang Du, Du, Qiang, Xin Yang Lu +4
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.2201.10097
arxiv created 2022/01/25 · openalex publication_date 2022/01/25 · arxiv updated 2022/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the minimization of an average distance functional defined on a two-dimensional domain Ω with an Euler elastica penalization associated with \pd Ω, the boundary of Ω. The average distance is given by ∫Ω \distp(x,\pd Ω)\d x where p≥ 1 is a given parameter, and \dist(x,\pd Ω) is the Hausdorff distance between \x\ and \pd Ω. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve \pd Ω, which is proportional to the integrated squared curvature defined on \pd Ω, as given by \la ∫\pd Ω κ\pd Ω2\d\H\llcorner \pd Ω1, where κ\pd Ω denotes the (signed) curvature of \pd Ω and \la>0 denotes a penalty constant. The domain Ω is allowed to vary among compact, convex sets of ℝ2 with Hausdorff dimension equal to 2\tcr. Under no a priori assumptions on the regularity of the boundary \pd Ω, we prove the existence of minimizers of Ep,\la. Moreover, we establish the C1,1-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.