What does this notation
$||\nabla u|| _p$ mean? I would like an exact definition.
Also I have seen $|\nabla u|_1$. Does this mean the $\sum_i|\partial_i u|$?
What does this notation
$||\nabla u|| _p$ mean? I would like an exact definition.
Also I have seen $|\nabla u|_1$. Does this mean the $\sum_i|\partial_i u|$?
Usually $\lVert \nabla u \rVert_p^p = \sum_{i=1}^n\lVert D_iu \rVert_p^p$ where $D_i$ means the $i$th partial derivative. The $\lVert \cdot \rVert_p$ refers to the $L^p$ norm, i.e., $\lVert u \rVert_p = (\int |u|^p)^{\frac{1}{p}}.$
For your second question: set $p =1$, so we have the $L^1$ norm.
If $u:\mathbb{R}^n\to\mathbb{R}$ is a function, then $\nabla u=(\partial_1 u,\ldots,\partial_n u)\in\mathbb{R}^n$ is the vector of partial derivatives. For any vector $x$, the $p$-norm $\left\|\cdot\right\|_p$ is defined as $ \left\|x\right\|_p = \left(\sum_{i}\left|x_i\right|^p\right)^{1/p}. $ In particular, $ \left\|\nabla u\right\|_p = \left(\sum_{i}\left|\partial_i u\right|^p\right)^{1/p} $ and $ \left\|\nabla u\right\|_1= \sum_{i}\left|\partial_i u\right|. $