Strong density for higher order Sobolev spaces into compact manifolds

Abstract

Given a compact manifold $N^n$, an integer $k\in {\mathbb{N}}_{\ast }$ and an exponent $1\le p<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k,p}(Q^m;N^n)$ when the homotopy group ${\pi }_{\lfloor kp\rfloor }(N^n)$ of order $\lfloor kp\rfloor$ is trivial. We also prove the density of maps that are smooth except for a set of dimension $m-\lfloor kp\rfloor -1$, without any restriction on the homotopy group of $N^n$

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• Density of bounded maps in Sobolev spaces into complete manifolds
• Uniform boundedness principles for Sobolev maps into manifolds
• Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes
• Higher order weak differentiability and Sobolev spaces between manifolds
• Asymptotic behavior of minimizing $p$-harmonic maps when $p\nearrow 2$ in dimension 2
• Weak approximation by bounded Sobolev maps with values into complete manifolds
• Quantitative characterization of traces of Sobolev maps
• An improved dense class in Sobolev spaces to manifolds