A nonlinear Korn inequality on a surface

Let ω be a domain in ${\mathbb{R}}^2$ and let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface $\mathit{\theta }(\overline{\omega })$”, asserting that, under ad hoc assumptions, the ${\mathit{H}}^1(\omega )$-distance between the surface $\mathit{\theta }(\overline{\omega })$ and a deformed surface is “controlled” by the ${\mathit{L}}^1(\omega )$-distance between their fundamental forms. Naturally, the ${\mathit{H}}^1(\omega )$-distance between the two surfaces is only measured up to proper isometries of ${\mathbb{R}}^3$.This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ${\mathit{\theta }}^k:\omega \to {\mathbb{R}}^3$, $k⩾1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, stay uniformly away from zero; and finally, the fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the fundamental forms of the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$.Such results have potential applications to nonlinear shell theory, the surface $\mathit{\theta }(\overline{\omega })$ being then the middle surface of the reference configuration of a nonlinearly elastic shell.