A spectral problem for the Laplacian in joined thin films

We consider a 3d multi-structure composed of two joined perpendicular thin films: a vertical one with small thickness h(n)(a) and a horizontal one with small thickness h(n)(b). We study the asymptotic behavior, as h(n)(a) and h(n)(b) tend to zero, of an eigenvalue problem for the Laplacian defined on this multi-structure. We shall prove that the limit problem depends on the value q = lim(n) h(n)(a/)h(n)(b). Precisely, we pinpoint three different limit regimes according to q belonging to ]0,+infinity 8[, q equal to +infinity, or q equal to 0. We identify the limit problems and we also obtain H-1-strong convergence results.

Automated summary

This article investigates a 3D multi-structure consisting of two thin films joined perpendicularly: a vertical film with small thickness h(n)(a) and a horizontal film with small thickness h(n)(b). The asymptotic behavior of an eigenvalue problem for the Laplacian on this multi-structure is studied as h(n)(a) and h(n)(b) approach zero. The limit problem is shown to depend on the value q = lim(n) h(n)(a/)h(n)(b), and three different limit regimes are identified based on the value of q belonging to ]0,+infinity 8[, q equal to +infinity, or q equal to 0. The limit problems are identified and H-1-strong convergence results are obtained.