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Hi!

Let $M$ be a compact manifold possibly with boundary with $\dim(M)=m$, let $N$ be a non compact manifold with $\dim(N)=n$. Let me recall the definition of the sobolev space $W^{1,p}(M,N)$. Embedding isometrically $N$ in some $\mathbb{R}^K$ we can define $$W^{1,p}(M,N):=\lbrace f\in W^{1,p}(M,\mathbb{R}^K)\textrm{ s.t. } f(p)\in N\textrm{ for a.e. }p\in M \rbrace$$ what can be said about the density of $C^{\infty}(M,N)$ in $W^{1,p}(M,N)$?

I'm particularly interested in the case $M=\overline{\Delta}$ ($\overline{\Delta}$ is the 2-dimensional closed unit disk) and $p=2$

Thank you in advance.

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  • $\begingroup$ Just to check -- it sounds like you know about the analogous results when $N$ is compact? For instance: (1) Schoen-Uhlenbeck intlpress.com/JDG/archive/1983/18-2-253.pdf, where it's proved that $\mathcal{C}^\infty(M, N)$ is dense in $\mathcal{W}^{1,2}(M, N)$ for $M$ compact 2-diml possibly with boundary and $N$ compact; and (2) Bethuel kryakin.com/files/Acta_Mat_%282_55%29/acta197_151/167/167_4.pdf, where the $p<m$ case is studied. $\endgroup$
    – macbeth
    Mar 4, 2011 at 16:00
  • $\begingroup$ yes, i'd like to know if there is an analogous of the density theorem of Schoen-Uhlenbeck (i.e. $p\geq\dim(M)$) for $N$ not compact. $\endgroup$
    – Italo
    Mar 4, 2011 at 16:21

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