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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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  • $\begingroup$ This seems to be a "geometric analysis" question: If I remember correctly, one generally uses classical vector calculus theorems to define such geometric weak derivatives. For the gradient, this should be Green's first identity: As we want the boundary terms to vanish, we may set $\int_{\Omega} u \Delta \phi = - \int_\Omega v \cdot \nabla \phi$ for all test functions $\phi$. Here the dot denotes the standard Euclidean inner product (so your definition depends on that choice). That is, the weak gradient $v$ of $u$ exists, if the above identity holds for some locally integrable vector field. $\endgroup$
    – user510186
    Feb 7, 2022 at 20:37
  • $\begingroup$ Of course the identity should be read as $\int_\Omega u \Delta \phi \, \operatorname{d}^3 x = - \int_\Omega v \cdot \nabla \phi \operatorname{d}^3x$ and the set $\Omega$ needs to be open (as otherwise the boundary terms cannot be assumed to vanish for compactly supported $\phi$). $\endgroup$
    – user510186
    Feb 7, 2022 at 21:43
  • $\begingroup$ The right approach seems to be to just define those operators using the ordinary notion of weak derivative and then look at what is needed to satisfy certain integral theorems in this instance (e.g. trace operators, etc.). But I cannot give you a satisfying answer to the latter, so I have deleted my previous response. $\endgroup$
    – user510186
    Feb 7, 2022 at 22:25

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