The Boundary Value Problem for Elliptic Equation in the Corner Domain

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Abstract


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.

E E Perepelkin

Joint Institute for Nuclear Research

Email: pevgeny@mail.ru

R V Polyakova

Joint Institute for Nuclear Research

Email: polykovarv@mail.ru

I P Yudin

Joint Institute for Nuclear Research

Email: yudin@jinr.ru

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Copyright (c) 2014 Перепёлкин Е.Е., Полякова Р.В., Юдин И.П.

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