A spectral problem for the Laplacian in joined thin films

Abstract

AbstractWe consider a 3d multi-structure composed of two joined perpendicular thin films: a vertical one with small thickness $$h^a_n$$ h n a and a horizontal one with small thickness $$h^b_n$$ h n b . We study the asymptotic behavior, as $$h^a_n$$ h n a and $$h^b_n$$ 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 $$q=\displaystyle {\lim _n\dfrac{h^b_n}{h^a_n}.}$$ q = lim n h n b h n a . Precisely, we pinpoint three different limit regimes according to q belonging to $$]0,+\infty [$$ ] 0 , + ∞ [ , q equal to $$+\infty $$ + ∞ , or q equal to 0. We identify the limit problems and we also obtain $$H^1$$ H 1 -strong convergence results.