Abstract

Let Ω⊂Rn\Omega \subset {\mathbb R}^n be a bounded domain with Lipschitz boundary, and assume that f:Ω×Rm×n→Rf: \Omega \times {\mathbb R}^{m \times n} \to {\mathbb R} is a Carathéodory integrand such that f(x,⋅)f(x, \cdot) is polyconvex for Ln{\mathcal L}^n- a.e. x∈Ωx \in \Omega. In this paper we consider integral functionals of the form F(u,Ω):=∫Ωf(x,Du(x)) dx,{\mathcal F}(u, \Omega):= \int_{\Omega} f(x, Du(x)) \, dx, where ff satisfies a growth condition of the type ∣f(x,A)∣≤c(1+∣A∣p),|f(x,A)| \le c (1 + |A|^p), for some c>0c>0 and 1≤p<∞1 \le p < \infty, and uu lies in the Sobolev space of vector-valued functions W1,p(Ω,Rm)W^{1,p}(\Omega, {\mathbb R}^m). We study the implications of a function u0u_0 being a critical point of F{\mathcal F}. In this regard we show among other things that if ff does not depend on the spatial variable xx, then every piecewise affine critical point of F{\mathcal F} is a global minimizer subject to its own boundary condition. Moreover for the general case, we construct an example exhibiting that the uniform positivity of the second variation at a critical point is not sufficient for it to be a strong local minimizer. In this example ff is discontinuous in xx but smooth in AA

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Ali Taheri

Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany

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Suggested citation

A. Taheri. “On Critical Points of Functionals with Polyconvex Integrands.” Journal of Convex Analysis 9 (2002), No. 1, 55–72. https://doi.org/10.68381/jca09003

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.