Reaction–diffusion problem in a thin domain with oscillating boundary and varying order of thickness

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

“…The analysis of eigenvalue problems for differential operators on thin domains has attracted noticeable interest in recent years, see, for example, [4–6, 11, 13, 15, 16, 23, 27, 35, 36, 42, 44, 46] and references therein. A somewhat complementary point of view is adopted in the asymptotic analysis of domains with small holes or perforations, see, for example, [1, 19, 22, 39, 45].…”

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

“…The analysis of eigenvalue problems for differential operators on thin domains has attracted noticeable interest in recent years, see e.g., [4,5,6,9,11,12,13,20,23,29,30,36,38,40] and references therein. A somehow complementary point of view is adopted in the asymptotic analysis of domains with small holes or perforations, see e.g., [1,16,19,33,39].…”

The p-Laplacian in thin channels with locally periodic roughness and different scales*

“…where g is a positive, bounded and periodic function satisfying some regularity hypothesis and ε > 0 is a small parameter which goes to zero. Thereby, in the limit ε → 0, the open set Q ε degenerates to the unit interval presenting oscillatory behaviour on the upper boundary (see for instance [1,3,5,[18][19][20][21][22] where similar approach are performed). The periodic rough boundary considered above is certainly a first step, but usually not enough, since most of the irregularities present in real applications are not periodic.…”