The interplay between two Euler–Lagrange operators relating to the nonlinear elliptic system $$\Sigma [(u, {\mathscr {P}}), \varOmega ]$$

We establish the existence of multiple whirling solutions to a class of nonlinear elliptic systems in variational form subject to pointwise gradient constraint and pure Dirichlet type boundary conditions. A reduced system for certain $$\mathbf{SO}(n)$$ SO ( n ) -valued matrix fields, a description of its solutions via Lie exponentials, a structure theorem for multi-dimensional curl free vector fields and a remarkable explicit relation between two Euler–Lagrange operators of constrained and unconstrained types are the underlying tools and ideas in proving the main result.

On a question of D. Serre

In this paper we give a negative answer to the question posed in D. Serre (Ann. Inst. Henri Poincaré C Anal. Non linéaire 35 (2018) 1209–1234, Open Question 2.1) about possible gains of integrability of determinants of divergence-free, non-negative definite matrix-fields. We also analyze the case in which the matrix-field is given by the Hessian of a convex function.

Nonlinear Donati compatibility conditions on a surface — Application to the intrinsic approach for Koiter’s model of a nonlinearly elastic shallow shell

An intrinsic approach to a mathematical model of a linearly or nonlinearly elastic body consists in considering the strain measures found in the energy of this model as the sole unknowns, instead of the displacement field in the classical approach. Such an approach thus provides a direct computation of the stresses by means of the constitutive equation. The main problem therefore consists in identifying specific compatibility conditions that these new unknowns, which are now matrix fields with components in [Formula: see text], should satisfy in order that they correspond to an actual displacement field. Such compatibility conditions are either of Saint-Venant type, in which case they take the form of partial differential equations, or of Donati type, in which case they take the form of ortho- gonality relations against matrix fields that are divergence-free. The main objective of this paper consists in showing how an intrinsic approach can be successfully applied to the well-known Koiter’s model of a nonlinearly elastic shallow shell, thus providing the first instance (at least to the authors’ best knowledge) of a mathematical justification of this approach applied to a nonlinear shell model (“shallow” means that the absolute value of the Gaussian curvature of the middle surface of the shell is “uniformly small enough”). More specifically, we first identify and justify compatibility conditions of Donati type guaranteeing that the nonlinear strain measures found in Koiter’s model correspond to an actual displacement field. Second, we show that the associated intrinsic energy attains its minimum over a set of matrix fields that satisfy these Donati compatibility conditions, thus providing an existence theorem for the intrinsic approach; the proof relies in particular on an interesting per se nonlinear Korn inequality on a surface. Incidentally, this existence result (once converted into an equivalent existence theorem for the classical displacement approach) constitutes a significant improvement over previously known existence theorems for Koiter’s model of a nonlinearly elastic shallow shell.

NONLINEAR SAINT–VENANT COMPATIBILITY CONDITIONS AND THE INTRINSIC APPROACH FOR NONLINEARLY ELASTIC PLATES

Let ω be a simply connected planar domain. First, we give necessary and sufficient nonlinear compatibility conditions of Saint–Venant type guaranteeing that, given two 2 × 2 symmetric matrix fields (Eαβ) and (Fαβ) with components in L2(ω), there exists a vector field (ηi) with components η1, η2 ∈ H1(ω) and η3 ∈ H2(ω) such that ½(∂αηβ + ∂βηα + ∂αη3∂βη3) = Eαβ and ∂αβη3 = Fαβ in ω for α, β = 1, 2. Second, we show that the classical approach to the Neumann problem for a nonlinearly elastic plate can be recast as a minimization problem in terms of the new unknowns Eαβ = ½(∂αηβ + ∂βηα + ∂αη3∂βη3) ∈ L2(ω) and Fαβ = ∂αβη3 ∈ L2(ω) and that this problem has a solution in a manifold of symmetric matrix fields (Eαβ) and (Fαβ) whose components Eαβ ∈ L2(ω) and Fαβ ∈ L2(ω) satisfy the nonlinear Saint–Venant compatibility conditions mentioned above. We also show that the analysis of such an "intrinsic approach" naturally leads to a new nonlinear Korn's inequality.