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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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2009 04 Nov 2009 10:37 05 Apr 2016 13:14 Springer 0003-9527 The original publication is available at www.springerlink.com https://doi.org/10.1007/s00205-008-0156-y http://www.ams.org/mathscinet-getitem?mr=2525114

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