Estimates in the Hardy-Sobolev space of the annulus and stability result

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space $H^{k, \infty}; k \in {\mathbb{N}}^*$ of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner: On the recovery of functions from pointwise boundary values in a Hardy-sobolev class of the disk. J.Comput.Apll.Math 46(1993), 255-69 and by S. Chaabane and I. Feki: Logarithmic stability estimates in Hardy-Sobolev spaces $H^{k,\infty}$. C.R. Acad. Sci. Paris, Ser. I 347(2009), 1001-1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.Comment: 14 pages. To be published in Czechoslovak Mathematical Journa

Effects of an external magnetic field on the gaps and quantum corrections in an ordered Heisenberg antiferromagnet with Dzyaloshinskii-Moriya anisotropy

We study the effects of external magnetic field on the properties of an ordered Heisenberg antiferromagnet with the Dzyaloshinskii-Moriya (DM) interaction. Using the spin-wave theory quantum correction to the energy, on-site magnetization, and uniform magnetization are calculated as a function of the field H and the DM anisotropy constant D. It is shown that the spin-wave excitations exhibit an unusual field-evolution of the gaps. This leads to various non-analytic dependencies of the quantum corrections on H and D. It is also demonstrated that, quite generally, the DM interaction suppresses quantum fluctuations, thus driving the system to a more classical ground state. Most of the discussion is devoted to the spin-S, two-dimensional square lattice antiferromagnet, whose S=1/2 case is closely realized in K2V3O8 where at H=0 the DM anisotropy is hidden by the easy-axis anisotropy but is revealed in a finite field. The theoretical results for the field-dependence of the spin-excitation gaps in this material are presented and the implications for other systems are discussed.Comment: 15+ pages, 14 Figure