We consider a 3d multi-structure composed of two joined perpendicular thin films: a vertical one with small thickness hᵃₙ h n a and a horizontal one with small thickness hᵇₙ h n b . We study the asymptotic behavior, as hᵃₙ h n a and hᵇₙ 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 ₙhᵇₙ/hᵃₙ. q = lim n h n b h n a . Precisely, we pinpoint three different limit regimes according to q belonging to ]0,+∞ [ ] 0 , + ∞ [ , q equal to +∞ + ∞ , or q equal to 0. We identify the limit problems and we also obtain H¹ H 1 -strong convergence results.
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Gaudiello et al. (2023) studied this question.