2015/03/02 by Seonghak Kim, Kim, Seonghak, Baisheng Yan +1 · 1 citation
Computer Science · Mathematics · #35D30 #35F60 #35K20 #35M13 #49K20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #math.AP #msc:35D30 #msc:35F60 #msc:35K20 #msc:35M13 #msc:49K20
paper · pdf · doi:10.48550/arxiv.1503.00772
arxiv created 2015/03/02 · openalex publication_date 2015/03/02 · arxiv updated 2015/03/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We prove that for all smooth nonconstant initial data the initial-Neumann boundary value problem for the Perona-Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona-Malik equation as a nonhomogeneous partial differential inclusion with linear constraint and uncontrollable components of gradient. We establish a general existence result by a suitable Baire's category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved.