A-B quasiconvexity and implicit partial differential equations

Abstract.

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

$\left\{ \begin{array}{ll} Bu(x) \in E & \quad \mbox{a.e.} x \in \Omega, u(x) = \varphi(x) &\quad \mbox{\rm for all} x \in \partial\Omega, \end{array} \right.$

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

$E(m):=\int_{\Omega }[\varphi (m)-\langle h_e;m\rangle ] dx+\frac{1}{2} \int_{\Bbb{R}^{3}}|h_{m}|^{2} dx$

where the magnetization $m:\Omega \to \Bbb{R}^{3}$ is taken with values on the unit sphere $S^{2}, h_{m}:\Bbb{R}^{3}\to \Bbb{R}^{3}$ 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 0 is on a face of $\partial \mathrm{co} Z$ or $0\in \mathrm{intco} Z$, where $\mathrm{co} Z$ denotes the convex hull of Z.