Differential and Integral Equations

Variational models for prestrained plates with Monge-Ampère constraint

Marta Lewicka, Pablo Ochoa, and Mohammad Reza Pakzad

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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

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

First available in Project Euclid: 23 June 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 74K20: Plates 74B20: Nonlinear elasticity


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

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