Fractional Sobolev isometric immersions of planar domains

2021/03/02 by Li, Siran, Pakzad, Mohammad Reza, Schikorra, Armin
#35D30 #46F10 #53A05 #53C24 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2103.01723

Abstract

We discuss C¹ regularity and developability of isometric immersions of flat domains into \mathbb R³ enjoying a local fractional Sobolev W^{1+s, \frac2s} regularity for 2/3 ≤ s< 1 , generalizing the known results on Sobolev and Hölder regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of W^2,\frac2s planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions.

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