This is one of my single author papers in the field of asymptotic analysis. It is concerned with the combined effect of two domain properties: thin geometry and perforation.
Thin Geometry:
Consider the Laplacian $\mathsf{-\Delta}$ on a domain $\mathsf{\Omega_\epsilon\subset\mathbb R^d}$ and suppose that $\mathsf{\Omega_\epsilon}$ approximates a graph $\mathsf\Gamma$ as $\mathsf{\epsilon\to0}$, as in the following figure.
One can prove1 that as $\mathsf{\epsilon\to 0}$ the Laplacian on $\mathsf{\Omega_\epsilon}$ converges to a Laplacian on the graph with certain matching conditions at the vertices (similar to the Kirchhoff conditions). The precise nature of these vertex conditions depends on the relative scaling of the edge neighbourhoods and the vertex neighbourhoods of the domain $\mathsf{\Omega_\epsilon}$.
Perforation:
A perforated domain is a domain, from which a periodic arrangement of balls is removed, where both the distance $\mathsf \epsilon$ and the radii $\mathsf{r_\epsilon}$ of the balls are much smaller than the diameter of the domain, as shown in the following figure (for more details, see my corresponding post).
This perforation results in a domain $\mathsf{\Omega_\epsilon}$, on which a partial differential equation can be studied (depending on the equation, this can be used to model materials with a fine microstructure).
It has been shown2 that
$$
\mathsf{-\Delta_{\Omega_\epsilon} \to -\Delta_{\Omega}+\mu} \quad \text{ as }\quad\mathsf{\epsilon\to 0},
$$
where $\mathsf\mu$ is a positive constant, which has been dubbed the “strange term”.2
Combination:
I my article, I studied the combined effect of thin, graph-like geometry and fine perforation, i.e. I studied domains which approximate a graph and in addition are finely perforated.
This combination leads to a striking observation: as in the case of purely thin geometry, a limit operator on the graph $\mathsf\Gamma$ can be found, which, on each edge of $\mathsf\Gamma$ is given by a shifted second derivative $\mathsf{-\frac{d^2}{dx^2}+\mu}$. On its own, this result would be only moderately surprising, but as it turns out the strange term $\mathsf\mu$ also appears in the vertex conditions of the limit problem! More precisely, the limit problem of the equation $\mathsf{(-\Delta+z)u=f}$ on $\mathsf{\Omega_\epsilon}$ turns out to be
$$
\begin{cases}
\mathsf{(-\Delta+z+\mu) u = f} &\textsf{ on }\mathsf \Gamma\\
\qquad \mathsf{\sum_{e\ni v} u'_e(v) = (z+\mu) \frac{|V|}{|\Omega_0|} u(v)}, &\textsf{ at each vertex }\mathsf v,
\end{cases}
$$
that is, the strange term $\mathsf\mu$ (which describes the “density” of the perforation) can be used as a tuning parameter to modify the Kirchhoff vertex conditions!
[P. Exner and O. Post. Convergence of spectra of graph-like thin manifolds. J. Geom. Phys. , 54(1) :77-115, 2005] ↩︎
[Cioranescu, Doina, and Francois Murat. A strange term coming from nowhere. Topics in the mathematical modelling of composite materials. Birkhäuser, Boston, MA, 1997. 45-93.] ↩︎
