On the Sobolev space of isometric immersions
Let \(\Omega\) be a bounded regular convex domain in \({\mathbb R}^2\) with Lipschitz boundary and consider the space of \(W^{2,2}\) local isometries from \(\Omega\) in the three-dimensional Euclidean space \[ I^{2,2}({\Omega}, {\mathbb R}^3) :=\{u\in W^{2,2}({\Omega}, {\mathbb R}^3); \nabla u\in O(2,3) \text{ a.e.}\} \] where \(O(2,3):=\{M\in M^{3\times 2}; M^TM=Id\}\). This space inherits the strong and weak topology of \(W^{2,2}({\Omega}, {\mathbb R}^3)\) and is closed under the weak convergence. The main purpose of this paper is to prove the following: Theorem I. {Smooth maps are strongly dense in \(I^{2,2}({\Omega}, {\mathbb R}^3)\). Equivalently, for any map \(f\in W^{1,2}({\Omega}, {\mathbb R}^2)\) with a.e. singular symmetric gradient there exists a sequence \(f_m\) of smooth maps satisfying the same condition and converging to \(f\) in \(W^{1,2}({\Omega}, {\mathbb R}^2)\). }
scientific article; zbMATH DE number 2135489
scientific article