Hinged and supported plates with corners Serguei A. Nazarov, Athanasios Stylianou and Guido Sweers Abstract. We consider the Kirchhoff-Love model for the supported plate, that is, the fourth order differential equation Δ 2 u = f with appropriate boundary conditions. Due to the expectation that a downwardly directed force f will imply that the plate, which is supported at its boundary, touches that support everywhere, one commonly identifies those boundary conditions with the ones for the so-called hinged plate: u =0=Δu - (1 - σ) κun. Engineers however are usually aware that rectangular roofs tend to bend upwards near the corners and this would mean that u = 0 is not appropriate. We will confirm this behaviour and show the difference of the supported and the hinged plates in case of domains with corners. Mathematics Subject Classification (2010). Primary: 35J86, 35J35; Secondary: 74K20. Keywords. supported plate, hinged plate, biharmonic operator, unilateral boundary conditions, variational inequality. 1. Introduction 1.1. Description and motivation Consider a thin plate subjected to a negative (downward) vertical load that lies freely at its sides on a supporting structure. Whenever the plate touches its support, the corresponding boundary condition fixes the height. A second boundary condition, which is necessary for the Kirchhoff-Love model of the plate in order to find a well-posed boundary value problem, comes naturally from the variational model describing the energy. This set of boundary conditions are known as hinged. However, if the plate does not touch its supporting structure, one finds a different set of boundary conditions. So it means that a supported plate may satisfy different sets of boundary conditions at different parts at the boundary. For a downward force one expects the plate to touch at least at some boundary parts. In the mathematical and engineering literature the (simply) supported and hinged boundary conditions are often confused; see also the comments by Blaauwendraad on [4, Chapter 13.4]. So before deriving a mathematical formulation let us clearly describe these two types of boundary conditions: • hinged: the deflection of the plate is zero on the boundary; • supported: the deflection of the plate cannot become negative on the boundary. So the main question is: Does a plate which is supported at its boundary by walls of constant height and is pushed downwards, touch this supporting structure everywhere? In engineering literature that considers supported plates, such as [4, 5], one finds that a rectan- gular supported plate will lift at the corners when pushed downwards. A rule of thumb is described by Figure 1. One approximates a thin plate by a configuration of 9 rigid tiles, elastically connected to each other, and supposes that the force is distributed over 12 points at the boundary. Pushed downwards by a uniformly distributed weight of size 1, the forces working on these 12 points act as