notation problem about $\left|\nabla f\right|$ and choice of vector norm

Given $f\in W^{1,p}(\mathbb{R}^N,\mathbb{R})$, I have seen a lot about this notation $|\nabla f|$ without specific pick of vector norm. Does the choice of a particular vector norm effect the equalities/ inequalities?

For example, the Coarea formula: $$\int_{\Omega} |\nabla f| = \int_{-\infty}^{\infty} H_{N-1} (f^{-1}(t))\mathrm{d}t.$$

And the total variation of a differentiable function (See the wiki Total variation) : for a $C^1(\bar{\Omega})$ function $f$, the total variation is \begin{align*} V(f,\Omega):=&\sup \left\{\int_{\Omega} f(x)\mathrm{div}\phi(x)\mathrm{d}x: \phi \in C_{c}^1(\Omega,\mathbb{R}^N), \|\phi\|_{\infty}\leqslant 1\right\}\\ =&\int_{\Omega}|\nabla f|\mathrm{d}x. \end{align*}

The total variation seems to pick $\ell_1$ or $\ell_2$ norm, since \begin{align} \int f\mathrm{div}\phi \leqslant \left|\int \phi \cdot \nabla f\right| &\leqslant |\phi|_{\infty}\int |\nabla f|_1\leqslant \int |\nabla f|_1\\ &\leqslant \int |\phi|_2\cdot |\nabla f|_2\leqslant \int |\nabla f|_2 \end{align}

However, by definition $V(f,\Omega)$ should be unique in its value.

Solution 1:

As stated in the comments, unless otherwise stated one usually assumes the $\ell_2$-norm. In the two examples you listed, this works for the following reasons.

Coarea formula: The coarea formula can be formulated for $f : \Bbb R^N \to \Bbb R$ Lipschitz as $$ \int_A J_f \,\mathrm dx = \int_{\Bbb R} \mathcal{H}^{N-1}(A \cap f^{-1}(y)) \,\mathrm dy $$ for any $A \subset \Bbb R^n$ measurable, where we define $$ J_f = \sqrt{\det(\nabla f \cdot \nabla f^T)}. $$ One can check that $J_f = |\nabla f|$ if we take the $\ell_2$-norm, and this does not hold in general with the $\ell_1$-norm.

TV formula: Observe that in the definition of $V(f,\Omega)$ we are using the definition $$ \lVert \phi \rVert_{\infty} = \mathrm{ess\,sup}_{x \in \Omega} |\phi(x)|,$$ where $\phi$ takes values in $\Bbb R^N.$ Taking the $\ell_2$-norm allows for the duality pairing $\nabla f(x) \cdot \phi(x) \leq |\nabla f(x)| |\phi(x)|,$ from which the second equality can be derived.

Note that you can take the $\ell_1$-norm for $|\nabla f|$ if you take the $\ell_{\infty}$-norm for $|\phi|$ instead, so the choice of $\ell_2$-norm is not as crucial here.