Liouville type results for local minimizers of the micromagnetic energy

International audienceWe study local minimizers of the micromagnetic energy in small ferromagnetic 3d convex particles for which we justify the Stoner-Wohlfarth approximation: given a uniformly convex shape $\Omega \subset {\mathbf{R}}^3$, there exist $\delta_c$>0 and $C > 0$ such that for $0 < \delta \leq \delta_c$ any \textit{local} minimizer $\mathbf{m}$ of the micromagnetic energy in the particle $\delta \Omega$ satisfies $\|\nabla \mathbf{m} \|_{L^2} \leqslant C \delta^2$.In the case of ellipsoidal particles, we strengthen this result by proving that, for $\delta$ small enough, \tmtextit{local} minimizers are exactly spatially uniform.This last result extends W.F. Brown's fundamental theorem for fine 3d ferromagnetic particles [Brown (1968), Di Fratta et al. (2011)] which states the same result but only for \textit{global} minimizers.As a by-product of the method that we use, we establish a new Liouville type result for locally minimizing $p$-harmonic maps with values into a closed subset of a Hilbert space. Namely, we establish that in a smooth uniformly convex domain of $\mathbf{R}^d$ any local minimizer of the $p$-Dirichlet energy ($p > 1$, $p \neq d$) is constant