Oberflächeninstabilitäten magnetischer Flüssigkeiten (Q798137)

The author continues a previous work about periodic equilibrium states of a magnetic fluid in an exterior magnetic field [Z. Angew. Math. Mech. 60, 235-2400 (1980; Zbl 0462.76044]. In this paper he calculates holohedry- invariant equilibrium solutions \(\eta\) on periodic hexagonal lattices \(\Lambda\). Giving the corresponding variation principle in terms of the energy E he looks for \(\Lambda\) -periodic solutions, which are elements of the Sobolev spaces \(H_ m\). They are shown to be analytic for \(m\geq 3\). For \(m\geq 4\), the first variation \(E_{\eta}\) is an analytic map between \(H_ m\) and \(H_{m-2}\). The equilibrium conditions are invariant with respect to the maximal subgroup G of the orthogonal group O(2). The on set of instability at a critical magnetic field \(H_{cr}\) is related to the eigenvalue zero of the second variation \(E_{\eta\eta }\), having the multiplicity six. Zero is only a nondegenerate eigenvalue on the subspaces \(H^ G_ m\) of G-invariant functions of \(H_ m\). Finally, approximate solutions of the scalar bifurcation equation are discussed in \(H^ G_ m\).