# Non-uniqueness of minimizers for strictly polyconvex functionals

Spadaro, E N (2009). Non-uniqueness of minimizers for strictly polyconvex functionals. Archiv for Rational Mechanics and Analysis, 193(3):659-678.

## Abstract

In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557–611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals,
$$\mathcal {F}(u)=\int_\Omega f(\nabla u(x)) {\rm d}x\quad{\rm and}\quad u\vert_{\partial\Omega}=u_0,$$
where Ω is homeomorphic to a ball.
We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surfaces.

## Abstract

In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557–611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals,
$$\mathcal {F}(u)=\int_\Omega f(\nabla u(x)) {\rm d}x\quad{\rm and}\quad u\vert_{\partial\Omega}=u_0,$$
where Ω is homeomorphic to a ball.
We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surfaces.

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## Additional indexing

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2009 04 Nov 2009 10:37 21 Sep 2018 11:36 Springer 0003-9527 The original publication is available at www.springerlink.com Green https://doi.org/10.1007/s00205-008-0156-y http://www.ams.org/mathscinet-getitem?mr=2525114

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