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This question is mostly to help me understand the idea behind the "weak curl", but I also hope to accomplish other objectives with this question/post as well, partially inspired from some of the "proof collecting" posts I've seen.

$1)$ I want to better understand the notion of "weak curl" with some examples.

$2)$ To hopefully discuss theorems surrounding when the weak versions of gradient, divergence, and curl (if possible), are equal to their strong counterparts and what this implies for solutions for PDEs.

$3)$ Collect illustrative examples of weak gradient, weak curl, and weak divergence in any number of dimensions or subsets. Maybe we can consider compact vs. non-compact subsets, upper/lower bounds on these quantities, disconnected spaces, and any related topic of interest.

WEAK GRADIENT:

let $\Omega\subset \mathbb{R}^n$, and let $u\in L^1_{loc}(\Omega)$ and $\phi\in C^{\infty}_c(\Omega)$. The function $v$ is called the "weak gradient" of $u$ if $\int_{\Omega}u\phi' d\mu=-\int_{\Omega}v\phi d\mu$. The "a-th" weak gradient is just

$\int_{\Omega}uD^{a}\phi d\mu=-(1)^a\int_{\Omega}v\phi d\mu$ $\forall \phi \in C^{\infty}_c(\Omega)$

WEAK DIVERGENCE:

v is called the "weak divergence" for $u\in L^2(\Omega)$ if we have $\int_{\Omega}u\phi d\mu=-\int_{\Omega}\langle v, \nabla \phi \rangle$ $\forall \phi \in C^{\infty}_c(\Omega)$

EDIT: I also realized I need clarification on the notation $\int_{\Omega}(v,\nabla \phi)$. I think this means we integrate w.r.t. each vector component of $\nabla \phi$, so for $\mathbb{R}^2$ we have $\int_{\mathbb{R}}\int_{\mathbb{R}}|v_1 \nabla \phi_1 v_2 \nabla \phi_2|^2 d\mu d\lambda$.

WEAK CURL:

This is a bit more delicate, and I am only aware of the definition where $\Omega\subset \mathbb{R}^3$. First, we need to define $u:\mathbb{R}^2\rightarrow \mathbb{R}^2$ with $u=(u_0, u_1)$ where $u_0\in L^2(\Omega)$, and $u_1\in L^2(\partial \Omega)$. Let $n$ be a normal vector to the boundary $\partial \Omega$.

$v$ is called the "weak curl" if we have: $v=curl(u)=\langle u_0, (\nabla \times \phi) \rangle + \langle u_1 \times n, \phi \rangle $ $\forall \phi \in C^{\infty}_c(\Omega)$ where the inner product here is the standard $L^2$ inner product.

Any interesting remarks/theorems are also welcome.

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