## Differential and Integral Equations

### Variational models for prestrained plates with Monge-Ampère constraint

#### Abstract

We derive a new variational model in the description of prestrained elastic thin films. The model consists of minimizing a biharmonic energy of the out-of plane displacements $v\in W^{2,2}(\Omega, \mathbb{R})$, satisfying the Monge-Ampèere constraint: $$\det\nabla^2v = f .$$ Here, $f=-\mbox{curl}^T\mbox{curl} (S_g)_{2\times 2}$ is the linearized Gauss curvature of the incompatibility (prestrain) family of Riemannian metrics $G^h= \mbox{Id}_3 + 2 h^\gamma S_g+ h.o.t.$, imposed on the referential configurations of the thin films with midplate $\Omega$ and small thickness $h$. We further discuss multiplicity properties of the minimizers of this model in some special cases.

#### Article information

Source
Differential Integral Equations, Volume 28, Number 9/10 (2015), 861-898.

Dates
First available in Project Euclid: 23 June 2015

https://projecteuclid.org/euclid.die/1435064543

Mathematical Reviews number (MathSciNet)
MR3360723

Zentralblatt MATH identifier
1363.74063

Subjects
Primary: 74K20: Plates 74B20: Nonlinear elasticity

#### Citation

Lewicka, Marta; Ochoa, Pablo; Pakzad, Mohammad Reza. Variational models for prestrained plates with Monge-Ampère constraint. Differential Integral Equations 28 (2015), no. 9/10, 861--898. https://projecteuclid.org/euclid.die/1435064543