I have two domains $A$ and $B$, they are defined such that the function $f(\mathbf{x})$ takes the value $x_A$ in $A$ and $x_B$ in B. $f$ is $C^{\infty}$
There exists a sharp interface of very small size $\epsilon$ determined by the function $G(d(\mathbf{x}))=f(\mathbf{x})$ between the two domains. $G$ is $C^{\infty}$.
$d(\mathbf{x})$ is the signed distance from the interface between the 2 domains. It has to be noted that $\nabla d(\mathbf{x})=\mathbf{n}$ where $\mathbf{n}$ is the unit vector, normal to this interface, whereas $\partial_{ij}d=Q_{ij}$ the partial derivatives with regards to the dimensions $i$ and $j$ is the curvature tensor. Let $G$ be positive for $\mathbf{x} \in A$ and negative for $\mathbf{x} \in B$.
I would like to build the Dirac delta corresponding to this interface in function of the function $G$ and its derivatives, such that : $$ \int_{A+B}F \delta(\mathbf{x})dV=\int_{\partial A} FdS$$ where $F$ is any function. The first integral is on the whole volume and the r.h.s integral is taken just on the surface of the domain $A$.
I suspect that this has been done somewhere since it looks quite fundamental but I did not find it.