Sobolev approximation lifts to $L^p$ convergence of the exterior powers

Question

I am reading the book "Geometric Function Theory and Non-linear Analysis", where the following claim is used:

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. Let $f \in W^{1,s}(\Omega,\mathbb{R}^n)$, and let $f_n \in C^{\infty}(\Omega,\mathbb{R}^n)$ converge to $f$ in $W^{1,s}$. Suppose $s \ge k \in \mathbb{N}$.

Then $\bigwedge^k df_n $ converges to $\bigwedge^k df $ in $L^1$, i.e $$ \int_{\Omega}|\bigwedge^k df_n -\bigwedge^k df | \to 0,$$ where the norm $|\cdot|$ is the standard Euclidean norm on linear maps between exterior powers $\Lambda_k(\mathbb{R}^n) \to \Lambda_k(\mathbb{R}^n)$.

Question: How to prove this claim?

My naive approach was to guess that $$ |\bigwedge^k df_n -\bigwedge^k df | \le C |df_n-df|^k, \tag{1}$$ which would imply

$$ ||\bigwedge^k df_n -\bigwedge^k df ||_ 1 \le C||df_n-df||_k^k \le \tilde C||df_n-df||_s^k \to 0.$$

However, estimate $(1)$ is false.

Edit: I tried to prove this via induction on $k$. However, I hit an obstacle.

Answer 1

This type of estimate is proved in Section 13.3 (1st ed), "Jacobians and wedge products revisited" by applying the Hadamard-Schwarz inequality (Section 9.9), and Hölder's inequality.

We can use the crude bound $|\wedge^k f-\wedge^k f_n|\leq \sum_{i_1<\dots<i_k} |(\wedge^k f-\wedge^k f_n)(e_{i_1}\wedge\dots\wedge e_{i_k})|$ to reduce to a problem about multivectors $\omega_1,\dots,\omega_k,\omega'_1,\dots,\omega_k'.$ We apply a telescoping sum and triangle inequality to bound $$\int|(\omega_1\wedge\dots\wedge\omega_k)-(\omega'_1\wedge\dots\wedge\omega'_k)|$$ by $k$ integrals of the form $$\int|\omega_1\wedge\dots\wedge\omega_{i-1}\wedge(\omega_i-\omega'_i)\wedge\omega'_{i+1}\wedge\dots\wedge\omega'_k|.$$ By Hadamard-Schwarz, this is bounded by $$C_{n,k}\int|\omega_1|\dots |\omega_{i-1}|\cdot|\omega_i-\omega'_i|\cdot|\omega'_{i+1}|\cdots|\omega'_n|,$$ which by Hölder's inequality is bounded by $$C_{n,k}\|\omega_1\|_k\dots \|\omega_{i-1}\|_k\cdot\|\omega_i-\omega'_i\|_k\cdot\|\omega'_{i+1}\|_k\cdots\|\omega'_{n}\|_k$$ which tends to zero if each $\omega'_i$ tends to $\omega_i$ in $L^k.$

Answer 2

This is a partial attempt at a proof. (At first I thought it was a complete proof, but now I see a gap; details are below).

We use induction on $k \le s$. That is, we think of $s$ as fixed, and let $k=1,\dots,[s]$.

The base case $(k=1)$:

$$ || df_n - df ||_ 1 \le C||df_n-df||_s \to 0. $$

The inductive step:

Suppose the claim holds for $k$. Since for the linear map $\bigwedge^k A:\Lambda_k(\mathbb{R}^n) \to \Lambda_k(\mathbb{R}^n) $, its norm is given by $$ |\bigwedge^k A|^2=\sum_{1 \le i_1 < \dots < i_k \le n} |\bigwedge^kA (e_{i_1} \wedge \dots \wedge e_{i_k}) |^2=\sum_{1 \le i_1 < \dots < i_k \le n} |A e_{i_1} \wedge \dots \wedge Ae_{i_k} |^2,$$

it suffices to prove

$$ || (\bigwedge^{k+1} df_n-\bigwedge^{k+1} df)(e_{i_1} \wedge \dots \wedge e_{i_{k+1}})||_1 \to 0, \tag{1}$$

for a single multii-index $I=(i_1,\dots,i_{k+1})$.

Indeed, suppose we proved $(1)$. Then

$$ ||\bigwedge^{k+1} df_n -\bigwedge^{k+1} df ||_1 =\int_{\Omega} \big( \sum_I |(\bigwedge^{k+1} df_n-\bigwedge^{k+1} df)(e_{i_1} \wedge \dots \wedge e_{i_{k+1}})|^2 \big)^{\frac{1}{2}} \le $$ $$\int_{\Omega} \sum_I | (\bigwedge^{k+1} df_n-\bigwedge^{k+1} df)(e_{i_1} \wedge \dots \wedge e_{i_{k+1}})|=\sum_I \int_{\Omega} | (\bigwedge^{k+1} df_n-\bigwedge^{k+1} df)(e_{i_1} \wedge \dots \wedge e_{i_{k+1}})|=$$ $$ \sum_I || (\bigwedge^{k+1} df_n-\bigwedge^{k+1} df)(e_{i_1} \wedge \dots \wedge e_{i_{k+1}})||_1.$$

We prove $(1)$:

$$ | (\bigwedge^{k+1} df_n-\bigwedge^{k+1} df)(e_{i_1} \wedge \dots \wedge e_{i_{k+1}})| \le $$ $$| df_n e_{i_1 }\wedge \dots df_n e_{i_{k} } \wedge df_n e_{i_{k+1}}-df_n e_{i_1 }\wedge \dots df_n e_{i_{k} } \wedge df e_{i_{k+1}} | +| df_n e_{i_1 }\wedge \dots df_n e_{i_{k} } \wedge df e_{i_{k+1}} -df e_{i_1 }\wedge \dots df e_{i_{k} } \wedge df e_{i_{k+1}}| =| df_n e_{i_1 }\wedge \dots df_n e_{i_{k} } \wedge (df_n e_{i_{k+1}}- df e_{i_{k+1}}) | +| (df_n e_{i_1 }\wedge \dots df_n e_{i_{k} } -df e_{i_1 }\wedge \dots df e_{i_{k} } )\wedge df e_{i_{k+1}} |\le$$

$$ C|\bigwedge^k df_n (e_{i_1} \wedge \dots \wedge e_{i_{k}})|\cdot |(df_n-df)e_{i_{k+1}}|+C|(\bigwedge^k df_n -\bigwedge^k df)(e_{i_1} \wedge \dots \wedge e_{i_{k}})| \cdot |df e_{i_{k+1}}|\le$$

$$ C \big( |\bigwedge^k df_n||df_n-df| +|\bigwedge^k df_n -\bigwedge^k df| |df | \big).$$

So, up to a multiplicative constant, $|| (\bigwedge^{k+1} df_n-\bigwedge^{k+1} df)(e_{i_1} \wedge \dots \wedge e_{i_{k+1}})||_1$ is dominated by $$ \int_{\Omega} |\bigwedge^k df_n||df_n-df| +|\bigwedge^k df_n -\bigwedge^k df| |df | \tag{2}$$

Analysing the first summand, we get $$ \int_{\Omega} |\bigwedge^k df_n||df_n-df| \le C \int_{\Omega} | df_n|^k|df_n-df| \le C || | df_n|^k ||_{\frac{k+1}{k}}||df_n-df||_{k+1} ,$$ where we used Holder inequality for $p=\frac{k+1}{k},q=k+1$. (Note that by assumption, $df_n \in L^{k+1}$ , so $| df_n|^k \in L^{\frac{k+1}{k}} $ ).

Thus,

$$ \int_{\Omega} |\bigwedge^k df_n||df_n-df| \le C ||df_n||_{k+1}^k \cdot ||df_n-df||_{k+1} \to C ||df||_{k+1}^k \cdot 0 =0.$$

The problem is what to do with the second summand.

Here we are supposed to use our induction hypothesis, but it seems that to show

$$|| \, \, |\bigwedge^k df_n -\bigwedge^k df| |df | \,\, ||_1 \to 0,$$ we need to know $||\, \, |\bigwedge^k df_n -\bigwedge^k df|\, \, ||_{\frac{k+1}{k}} \to 0$, while the induction hypothesis only gives us $$ ||\, \, |\bigwedge^k df_n -\bigwedge^k df|\, \, ||_1 \to 0. $$

Perhaps we should strengthen the induction hypothesis somehow?