Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with Lipschitz boundary $\partial\Omega$ and outward unit normal $n$.
I want to study the characterize whether a vector function defined on $\Omega$ admits a scalar potential or a vector potential.
Let's introduce the following spaces of harmonic fields $$ H(m;\Omega)=\{v\in (L^2(\Omega))^3:curl(v)=0\,\,\text{in}\,\Omega, div(v)=0\,\,\text{in}\,\Omega, v\cdot n =0\,\,\text{on}\,\partial\Omega\} $$
$$ H(e;\Omega)=\{v\in (L^2(\Omega))^3:curl(v)=0\,\,\text{in}\,\Omega, div(v)=0\,\,\text{in}\,\Omega, v\times n =0\,\,\text{on}\,\partial\Omega\} $$
The following decompositions hold.
Theorem 1: Any $v\in (L^2(\Omega))^3$ can be decomposed into $$ v=curl(Q)+\nabla\psi+\rho $$ where - each term is orthogonal to the others - if $curl(v)=0$ in $\Omega$, then $Q=0$ in $\Omega$ - if $div(v)=0$ in $\Omega$ and $v\cdot n=0$ on $\partial\Omega$, then $\nabla\psi=0$ in $\Omega$ - if $v\perp H(m;\Omega)$, then $\rho=0$ in $\Omega$ - $Q$ is solution to $$ \begin{cases} curl curl (Q)= curl(v) & \text{in } \Omega \\ div(Q)=0 & \text{in } \Omega\\ Q\times n=0 & \text{on } \partial\Omega\\ Q\perp H(e;\Omega) \end{cases} $$ - $\psi$ is solution to $$ \begin{cases} \Delta\psi= div(v) & \text{in } \Omega \\ \nabla\psi\cdot n= v\cdot n & \text{on } \partial\Omega \end{cases} $$ - $\rho$ is the $L^2$-orthogonal projection of $v$ onto $H(m;\Omega)$
Theorem 2: Any $v\in (L^2(\Omega))^3$ can be decomposed into $$ v=curl(A)+\nabla\phi+\eta $$ where - each term is orthogonal to the others - if $curl(v)=0$ in $\Omega$ and $v\times n=0$ on $\partial\Omega$, then $A=0$ in $\Omega$ - if $div(v)=0$ in $\Omega$, then $\nabla\phi=0$ in $\Omega$ - if $v\perp H(e;\Omega)$, then $\eta=0$ in $\Omega$ - $A$ is solution to $$ \begin{cases} curl curl (A)= curl(v) & \text{in } \Omega \\ div(A)=0 & \text{in } \Omega\\ A\cdot n = 0 & \text{in } \Omega\\ A\times n=v\times n & \text{on } \partial\Omega\\ Q\perp H(m;\Omega) \end{cases} $$ - $\phi$ is solution to $$ \begin{cases} \Delta\phi= div(v) & \text{in } \Omega \\ \phi=0 & \text{on } \partial\Omega \end{cases} $$ - $\eta$ is the $L^2$-orthogonal projection of $v$ onto $H(e;\Omega)$
Now, the following theorems follow easily.
Theorem 3: Let $v\in (L^2(\Omega))^3.$ There exists a scalar potential $\psi$ if and only if $curl(v)=0$ in $\Omega$ and $v\perp H(m;\Omega)$.
Theorem 4: Let $v\in (L^2(\Omega))^3.$ There exists a vector potential $A$ if and only if $div(v)=0$ in $\Omega$ and $v\perp H(e;\Omega)$.
From the first two theorems it's really immediate to deduce the latter two. My problem is to remember the first the two results. Do you have any suggestions? Why $Q$, $\phi$, $A$ and $\phi$ are defined like that?
(I'm looking for a intuitive/heuristic motivation)