question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$

2011-01-07T15:18:18Z

Hi, i'm reading the proof of the fact that $C^{\infty}(M,N)$ is dense in the sobolev space $W^{1,m}(M,N)$, where $M,N$ are compact riemannian manifolds of dimension respectively $m,n$. I recall quickly the definition of $W^{1,m}(M,N)$. Embedding $N$ isometrically in a $\mathbb{R}^J$ we define $$W^{1,m}(M,N)= \left\{ f \in W^{1,m}(M,\mathbb{R}^J)\quad \mid \quad f(x)\in N\quad \textrm{for a.e.}\quad x\in M \right\} $$ The proof goes as follows: embed $M$ isometrically in $\mathbb{R}^K$, consider a tubular neighborhood $T_M$ sufficiently small to have a smooth "nearest point" projection $\pi_M:T_M\to M$. Let $f\in W^{1,m}(M,N)$ consider $F=f\circ\pi_M$ and then its mollifications $F_{\epsilon} = F \ast \phi_{\epsilon}$ . We want that values of $ F_{\epsilon}$ stay in a tubular neighborhood of $N$ sufficiently small to have a smooth "nearest point" projection $\pi_N:T_N\to N$. To do this we want to estimate $dist(F_{\epsilon}(x),N)$ and show that it goes to zero as $\epsilon$ goes to zero.

Now comes the part that i don't understand, the author says that follows from poincare inequality that $$\frac{1}{\mu(B(x,\epsilon))}\int_{B(x,\epsilon)}|F(y)-F_{\epsilon}(x)|^m dy\leq \frac{C\epsilon^m}{\mu(B(x,\epsilon))}\int_{B(x,\epsilon)}|\nabla F(y)|^m dy$$ With C a positive constant. But to have an inequality like this don't i need that the mean of $F(y)-F_{\epsilon}(x)$ is zero on $B(x,\epsilon)$?

This is the crucial estimate because then it is enough to consider $\pi_N\circ F_{\epsilon}$ and i get a sequence of smooth maps converging to $f$ in the sobolev norm. Please can anyone help? Or anyone does know another proof or reference to a proof of the density of $C^{\infty}(M,N)$ in $W^{1,m}(M,N)$?

Thank you in advance.

Answer by Deane Yang for question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$

2011-01-07T16:40:17Z

I don't have the time to work out a real answer to this question but estimates like this can usually be proved using the following:

$ F(y) - F_{\epsilon}(x) = \int \phi_\epsilon(x,z)(F(y) - F(z)) dz $

where $\phi_\epsilon$ is the mollifier kernel function and then substituting in

$ F(y) - F(z) = \int_0^1 \gamma'(t)\cdot\nabla F(\gamma(t)) dt $

where $\gamma$ is a constant speed geodesic joining $z$ to $y$.

You just need to estimate the $L_m$ norm of the right side of the first equation with the second equation substituted in. This should lead to something close to what you want.

question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$ - MathOverflow

question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$

Answer by Deane Yang for question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$