We prove the regularity of solutions to the strain tensor equation on degenerated hyperbolic surfaces $S$ where the Gauss curvature is zero on a part of the boundary. Furthermore, we obtain the density property that smooth infinitesimal isometries are dense in the W$^{2,2}({S},\mathbb{R}^3)$ infinitesimal isometries. Finally, the matching property is established. Those results are important tools in obtaining recovery sequences ($\Gamma$-lim sup inequality) for dimensionally reduced shell theories in elasticity. 

　　Publication: 

　　SIAM Journal on Mathematical Analysis, Vol. 54, Iss. 4 (2022)  

　　https://doi.org/10.1137/21M1440906 

　　Author: 

　　Liangbiao Chen 

　　Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China 

　　Pengfei Yao 

　　School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China 

　　Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People's Republic of China 

　　Email: pfyao@iss.ac.cn