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How can I find a functional that has a minimum satisfying the PDE $$ \begin{cases} \nabla\cdot \mathbf{B} = 0 & \text{on }\Omega\,,\\ \nabla\times \mathbf{B} = 0 & \text{on }\Omega\,,\\ \mathbf{B}\cdot\hat{\mathbf{n}}=0 & \text{on }\partial\Omega\, , \end{cases} $$ where $\hat{\mathbf{n}}$ is a vector normal to the surface of $\Omega$ and $\Omega\in\mathbb{R}^3$. In my case the domain $\Omega$ is such that there exists a vector field $\mathbf{A}$ such that $B = \nabla\times\mathbf{A}$, so the above PDE can be written as $$ \begin{cases} \nabla\times(\nabla\times \mathbf{A}) = 0 & \text{on }\Omega\,,\\ (\nabla\times \mathbf{A})\cdot\hat{\mathbf{n}}=0 & \text{on }\partial\Omega\, . \end{cases} $$ Note that there should be a nonzero solution to this PDE since my domain has the topology of a torus.

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The Hodge decomposition gives you that $B$ is harmonic on $\Omega$ and thus minimizes the vector Dirichlet energy $$E(v) = (\nabla \cdot v)^2 + \|\nabla \times v\|^2.$$ Of course, so does the zero vector field, so you will need to bar it by adding, say, a norm constraint: $$B = \underset{v}{\operatorname{argmin}} \int_{\Omega} \left[(\nabla \cdot v)^2 + \|\nabla \times v\|^2\right]\,dV \quad \mathrm{s.t.}\quad \begin{array}{c}\langle v, \hat{n}\rangle=0 \mathrm{\ on\ } \partial \Omega,\\ \int_{\Omega} \|v\|^2\,dV = 1.\end{array}$$

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  • $\begingroup$ Do you know of any books that study the Dirichlet energy of vector fields? I've seen the Dirichlet energy before, but for classical real valued harmonic functions that minimise $\int_\Omega||\nabla\phi||^2\,\text{d}V$. Also, if $B$ is a harmonic vector field, then does it satisfy some sort of maximum principle? $\endgroup$ – Vishnu Mangalath Sep 6 '18 at 3:26

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