The goal of this paper is to study operators sum of $p$-Laplacian type operators. We address the problems of existence and uniqueness of solutions, this last point leading to some challenging issues in the case of quasilinear combinations of such $p$-Laplacians.

In this paper we first introduce a new shell model that can be applied for all kinds of geometries of the middle surface of the shell. Then we show that our model is close to Koiter's nonlinear shell model in a specific sense. Finally, we establish the existence of a minimizer for this new model.

We estimate the constants appearing in Korn inequalities in terms of the norm of a linear and continuous inverse to the divergence operator defined on a same domain, and of a few scalar parameters modeling the shape of the domain.

One of Korn’s scaled inequalities for shells asserts that the $H^1$-norm of a displacement field of a shell with thickness $2ε$ clamped on a portion of its lateral boundary, once scaled to a domain independent of $ε,$ is bounded above by the $L^2$-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order $ε^{−1}.$ We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190].

The aim of the paper is to recall the importance of the study of invertibility and monotonicity of stress-strain relations for investigating the nonuniqueness and bifurcation of homogeneous solutions of the equilibrium problem of a hyperelastic cube subjected to equiaxial tensile forces. In other words, we reconsider a remarkable possibility in this nonlinear scenario: Does symmetric loading lead only to symmetric deformations or also to asymmetric deformations? If so, what can we say about monotonicity for these homogeneous solutions, a property which is less restrictive than the energetic stability criteria of homogeneous solutions for Rivlin’s cube problem. For the Neo-Hooke type materials we establish what properties the volumetric function $h$ depending on ${\rm det}F$ must have to ensure the existence of a unique radial solution (i.e. the cube must continue to remain a cube) for any magnitude of radial stress acting on the cube. The function $h$ proposed by Ciarlet and Geymonat satisfies these conditions. However, discontinuous equilibrium trajectories may occur, characterized by abruptly appearing non-symmetric deformations with increasing load, and a cube can instantaneously become a parallelepiped. Up to the load value for which the bifurcation in the radial solution is realized local monotonicity holds true. However, after exceeding this value, monotonicity no longer occurs on homogeneous deformations which, in turn, preserve the cube shape.