A-B Quasiconvexity and Implicit Partial Differential Equations



Bernard Dacorogna
Département de Mathématiques
EPFL
1015 Lausanne, Suisse
email: Bernard.Dacorogna@epfl.ch



and



Irene Fonseca Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
email: fonseca@andrew.cmu.edu



ABSTRACT: The study of existence of solutions of boundary-value problems for differential inclusions

\begin{eqnarray*}\left\{ \begin{array}{ll} Bu(x) \in E & \quad \text{a.e.} \, x ... ...\quad \text{for all} \, x \in \partial\Omega, \end{array}\right. \end{eqnarray*}


where $\varphi \in C^1_{\rm piec}(\overline{\Omega};\mathbb R^N)$, $\Omega$ is an open subset of $\mathbb R^n$, $E\subset \mathbb R^{m\times n}$ is a compact set, and B is a $m\times n$-valued first order differential operator, is undertaken. As an application, minima of the energy for large magnetic bodies

\begin{displaymath}E(m):=\int_{\Omega }[\varphi (m)-\langle h_{e};m\rangle ]\,dx+\frac{1}{2}% \int_{\Bbb{R}^{3}}\vert h_{m}\vert^{2}\,dx \end{displaymath}

where the magnetization $m:\Omega \to \Bbb{R}^{3}$ is taken with values on the unit sphere S2, $h_{m}:\Bbb{R}^{3}\to \Bbb{R}^{3}$ is the induced magnetic field satisfying $\mathrm{curl}\,h_{m}=0$ and $\mathrm{div}% \,(h_{m}+m\chi _{\Omega })=0$, $\varphi $ is the anisotropic energy density, and the applied external magnetic field is given by $h_{e}\in \Bbb{R}^{3}$, are fully characterized. Setting $Z:=\{\xi \in S^{2}:\psi (\xi )=\min_{m\in S^{2}}\psi (m)\}$ with $% \psi (m):=\varphi (m)-\langle h_{e};m\rangle $, it is shown that E admits a minimizer $m\in L^{\infty }$ with $h_{m}\equiv 0$ if and only if either 0is on a face of $\partial \mathrm{co}\,Z$ or $0\in \mathrm{intco}\,Z$, where $\mathrm{co}\,Z$ denotes the convex hull of Z.


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