Received: 20 February 2018 Revised: 20 February 2019 Accepted: 11 March 2019 DOI: 10.1002/mana.201800092 ORIGINAL PAPER Comparing variational methods for the hinged Kirchhoff plate with corners Colette De Coster 1 Serge Nicaise 1 Guido Sweers 2 1 Univ. Polytechnique Hauts-de-France, EA 4015 - LAMAV - FR CNRS 2956, F-59313 Valenciennes, France 2 Universität zu Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln, Deutschland Correspondence Serge Nicaise, Univ. Polytechnique Hauts-de- France, EA 4015 - LAMAV - FR CNRS 2956, F-59313 Valenciennes, France. Email: Serge.Nicaise@uphf.fr Abstract The hinged Kirchhoff plate model contains a fourth order elliptic differential equa- tion complemented with a zeroeth and a second order boundary condition. On domains with boundaries having corners the strong setting is not well-defined. We here allow boundaries consisting of piecewise  2,1 -curves connecting at corners. For such domains different variational settings will be discussed and compared. As was observed in the so-called Saponzhyan–Babushka paradox, domains with reentrant cor- ners need special care. In that case, a variational setting that corresponds to a second order system needs an augmented solution space in order to find a solution in the appropriate Sobolev-type space. KEYWORDS biharmonic operator, corner domains, hinged boundary condition MSC (2010) 35J35, 35J40, 74K20 1 INTRODUCTION The classical pde-formulation for the hinged Kirchhoff plate is the following boundary value problem: ⎧ ⎪ ⎨ ⎪ ⎩ Δ 2  =  in Ω,  =0 on Ω, Δ = (1 - )    on Ω, (1.1) with  the outward normal unit vector and where  is the signed curvature of the boundary Ω, positive on convex boundary sections and negative on concave parts (see [24]). The domain Ω ⊂ ℝ 2 resembles the plate in rest. The constant  is the Poisson ratio, a physical quantity depending on the material which satisfies  ∈ ( -1, 1 2 ] and usually lies near 3 10 . Only for some auxetic materials it is negative (see [13]). For a vertical force  ∶Ω → ℝ one is interested in a solution  ∶Ω → ℝ, which represents the deviation of the plate from the horizontal. A well-defined curvature  needs Ω to be  2 and then one may rewrite (1.1) as a coupled system: { -Δ =  in Ω, - = (1 - )    on Ω, and { -Δ =  in Ω,  =0 on Ω. (1.2) Mathematische Nachrichten. 2019;1–28. © 2019 WILEY-VCH Verlag GmbH &Co. KGaA, Weinheim 1 www.mn-journal.org