The Boundary Value Problem for Elliptic Equation in the Corner Domain

Modern accelerator systems and detectors contain magnetic systems of complex geometrical configuration. Design and optimization of the magnetic systems demand solving a nonlinear boundary-value problem of magnetostatic. The region in which the boundaryvalue problem is solved, consists of two sub-regions: a region of vacuum and a region of ferromagnetic. In view of the complex geometrical configuration of magnetic systems, the ferromagnetic/vacuum boundary can be nonsmooth, i.e. it contains a corner point near of which the boundary is formed by two smooth curves crossed in a corner point at some angle. For linear differential equations it is known that in such regions the solutions of the corresponding boundary-value problems can possess unlimitedly growing first derivatives near of the corner point. Some works consider a nonlinear differential equation of divergent type in the region with a corner and the opportunity of existence of solutions with unlimitedly growing module of gradient near the corner point is shown. The present work analyzes the region consisting of two sub-regions (ferromagnetic/vacuum) divided by a boundary with the corner point. In this region one considers a formulation of the magnetostatics problem with respect to two scalar potentials. Nonlinearity of the boundary-value problem is related to the function of magnetic permeability which depends upon the module of gradient of the solution to the boundary-value problem. In a case when the function of magnetic permeability at big fields satisfies certain conditions, in this work a theorem of limitation of the module of gradient of the solution near the corner point is proved.

magnet systems, mathematical modeling, boundary value problem, elliptic equations, the behavior of solutions in the corner domain