Boundary conditions for magnetization in magnetic nano-elements
  We show that the dynamic magnetization at the edges of a thin magnetic element with finite lateral size can be described by new effective boundary conditions that take into account inhomogeneous demagnetizing fields near the element edges. These fields play a dominant role in the effective pinning of the dynamic magnetization at the boundaries of mesoscopic and nano-sized magnetic elements. The derived effective boundary conditions generalize well-known Rado-Weertman boundary conditions and are reduced to them in the limiting case of a very thin magnetic element. 
K.Yu. Guslienko and A.N. Slavin