Weak surface divergence of a tangential field

Assume the surface-measure convention with ${\mathrm{A}\mathrm{C}}_{\omega }$ of the mathematical definition of the stationary surface distribution. Let $S\subset {\mathbb{R}}^3$ be a compact embedded $C^2$ hypersurface without manifold boundary, with a chosen continuous unit normal $n$ and its finite surface measure $dS$. At $x\in S$, define the tangent plane

$T_{x}S:=\{v\in {\mathbb{R}}^3:v\cdot n(x)=0\}\mathrm{.}$

For $\varphi \in C^1(S)$, meaning that its composition with each local $C^2$ surface chart is $C^1$, its surface gradient ${\mathrm{\nabla }}_S\varphi (x)\in T_{x}S$ is the unique vector satisfying

$v\cdot {\mathrm{\nabla }}_S\varphi (x)=d{\varphi }_x(v)\text{for every }v\in T_{x}S,$

where $d{\varphi }_x$ is the derivative of $\varphi$ in tangent directions. For a measurable tangential field $K:S\to {\mathbb{R}}^3$ with ${\int }_S\mathrm{\mid }K\mathrm{\mid }dS<\mathrm{\infty }$, define its weak surface divergence ${\mathrm{Div}}_{S}K$ as the linear functional on $C^1(S)$

$⟨{\mathrm{Div}}_{S}K,\varphi ⟩:=-{\int }_{S}K\cdot {\mathrm{\nabla }}_S\varphi dS\mathrm{.}$

This functional is well-defined because $S$ is compact, ${\mathrm{\nabla }}_S\varphi$ is continuous and bounded, and $K$ is integrable. If it has a continuous density $g:S\to \mathbb{R}$, meaning

$⟨{\mathrm{Div}}_{S}K,\varphi ⟩={\int }_{S}g\varphi dS\text{for every }\varphi \in C^1(S),$

then $g$ is called a continuous representative of the weak surface divergence. The definition itself does not assert that such a representative exists or is unique; a result using pointwise ${\mathrm{div}}_{S}K$ must establish both.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Surface distributions and interfacial jumps

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$