2019/11/04 by Guido De Philippis, De Philippis, Guido, Filip Rindler +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1911.01356
openalex publication_date 2019/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this survey we collect some recent results obtained by the authors and\ncollaborators concerning the fine structure of functions of bounded deformation\n(BD). These maps are \L1-functions with the property that the\nsymmetric part of their distributional derivative is representable as a bounded\n(matrix-valued) Radon measure. It has been known for a long time that for a\n(matrix-valued) Radon measure the property of being a symmetrized gradient can\nbe characterized by an under-determined second-order PDE system, the\nSaint-Venant compatibility conditions. This observation gives rise to a new\napproach to the fine properties of BD-maps via the theory of PDEs for measures,\nwhich complements and partially replaces classical arguments. Starting from\nelementary observations, here we elucidate the ellipticity arguments underlying\nthis recent progress and give an overview of the state of the art. We also\npresent some open problems.\n