2024/09/13 by Siran Li, Li, Siran
Engineering · #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Composite Material Mechanics #Differential Geometry (math.DG) #Elasticity and Material Modeling #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2409.08922
openalex publication_date 2024/09/13 · openalex created_date 2024/10/23 · openalex updated_date 2026/07/28
We survey recent developments on the analysis of Gauss--Codazzi--Ricci equations, the first-order PDE system arising from the classical problem of isometric immersions in differential geometry, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges.