Pointwise fixation along the edge of the Kirchhoff plate

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 distance $\varepsilon>0$ between them, and the second one is composed of points placed along a contour at distance $O(\varepsilon^{1+\alpha})$ from the edge. We prove that, in the case $\alpha\in[0,1/2)$, the limit passage as $\varepsilon\rightarrow+0$ leads to the plate rigidly clamped along the edge while, in the case $\alpha>1/2$, 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 $\alpha=1/2$ the boundary hinge-support conditions with friction occur in the limit. We discuss the available generalization of the results and open questions.