Density of bounded maps in Sobolev spaces into complete manifolds

Abstract

Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1,p}\cap L^{\infty })(Q^m;N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1,p}(Q^m;N^n)$ for $1\le p\le m$. The density always holds when $p$ is not an integer. When $p$ is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured for example by a uniform Lipschitz geometry of $N^n$. As a byproduct, we give necessary and sufficient conditions for the strong density of the set of smooth maps $C^{\infty }(\overline{Q^m};N^n)$ in $W^{1,p}(Q^m;N^n)$.

Accepted for publication in Annali di Matematica Pura ed Applicata (1923 -)

References in corpus (2)

• Strong density for higher order Sobolev spaces into compact manifolds
• Weak approximation by bounded Sobolev maps with values into complete manifolds

Cited by in corpus (5)

• Uniform boundedness principles for Sobolev maps into manifolds
• Asymptotic behavior of minimizing $p$-harmonic maps when $p\nearrow 2$ in dimension 2
• Existence of polyharmonic maps in critical dimensions
• Weak approximation by bounded Sobolev maps with values into complete manifolds
• Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group