Approximating $W^{2,2}$ isometric immersions

Hornung, Peter

Comptes Rendus. Mathématique, Tome 346 (2008) no. 3-4, pp. 189-192.

Soient $S\subset {\mathbb{R}}^2$ un domaine lipschitzien borné et $W_{\mathrm{iso}}^{2,2}(S;{\mathbb{R}}^3)$ l'ensemble $W_{\mathrm{iso}}^{2,2}(S;{\mathbb{R}}^3)=\{u\in W^{2,2}(S;{\mathbb{R}}^3):{(\mathrm{\nabla }u)}^T(\mathrm{\nabla }u)=\mathrm{Id}\text{p.p.}\}$. Sous une hypothèse supplémentaire de régularité sur la frontière ∂S (qui est satisfaite dans le cas où ∂S est continument différentiable par morceaux), nous prouvons que l'adhérence $W^{2,2}$ de $W_{\mathrm{iso}}^{2,2}(S;{\mathbb{R}}^3)\cap C^{\infty }(\overline{S};{\mathbb{R}}^3)$ est $W_{\mathrm{iso}}^{2,2}(S;{\mathbb{R}}^3)$.

Let $S\subset {\mathbb{R}}^2$ be a bounded Lipschitz domain and set $W_{\mathrm{iso}}^{2,2}(S;{\mathbb{R}}^3)=\{u\in W^{2,2}(S;{\mathbb{R}}^3):{(\mathrm{\nabla }u)}^T(\mathrm{\nabla }u)=\mathrm{Id}\text{a.e.}\}$. Under an additional regularity condition on the boundary ∂S (which is satisfied if it is piecewise continuously differentiable) we prove that the $W^{2,2}$ closure of $W_{\mathrm{iso}}^{2,2}(S;{\mathbb{R}}^3)\cap C^{\infty }(\overline{S};{\mathbb{R}}^3)$ agrees with $W_{\mathrm{iso}}^{2,2}(S;{\mathbb{R}}^3)$.

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