# Weak analogues of gradient, divergence, and curl (collecting examples)

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.

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.