Approximation of flat \(W ^{2,2}\) isometric immersions by smooth ones
Let \(S\subset {\mathbb R}^2\) be a bounded Lipschitz domain in the euclidean plane. On this domain consider the Sobolev space \(W_{\text{iso}}^{2,2}(S;{\mathbb R}^3)\) of Sobolev maps \(u:S\to {\mathbb R}^3\) into euclidean 3-space which are isometric immersions almost everywhere on \(S\). In the main theorem it is proved that the strong \(W^{2,2}\) closure of \(W_{\text{iso}}^{2,2}(S;{\mathbb R}^3)\cap C^{\infty}(\bar S; {\mathbb R}^3)\) agrees with \(W_{\text{iso}}^{2,2}(S;{\mathbb R}^3)\), i.e., the flat \(W^{2,2}\) isometric immersions of a planar Lipschitz domain into euclidean 3-space can be approximated by smooth ones. The result has applications in nonlinear elasticity where such immersions form the natural class of admissible deformations in Kirchhoff's plate theory.
scientific article; zbMATH DE number 5952962
scientific article