Abstract: We address the Sobolev-Neumann problem for the bi-harmonic equation describing the bending of the Kirchhoff plate with a traction-free edge but fixed at two rows of points. The first row is composed of points placed at the edge, at a small distance ? > 0 between them, and the second one is composed of points placed along a contour at distance O(?1+?) from the edge. We prove that in the case where ? ? [0, 1/2), the limit passage as ? ? +0 leads to the plate rigidly clamped along the edge while, in the case where ? > 1/2, under additional conditions, the limit boundary conditions become of the hinge support type. Based on the asymptotic analysis of the boundary layer in a similar problem, we predict that in the critical case ? = 1/2, the boundary hinge-support conditions with friction occur in the limit. We discuss the available generalization of the results and open questions.

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