Time harmonic solutions to a model of ferroelectric materials

Abstract

We discuss the model equations of ferroelectric media, represented by a bounded and regular domain

introduced by Greenberg in the case of time harmonic dependency of the solutions. The system reduces to a couple of nonlinear Maxwell's equations satisfied by the electromagnetic field ( H, E ) and the electric polarization vector P. By classical methods we prove the existence and uniqueness of the solutions for frequencies which are far from 0. We obtain the

-regularity of the solutions when the polarization satisfies the boundary condition P× n = 0. We also consider the particular 2-D case deduced from the full model by using the transverse electric (TE) and transverse magnetic (TM) approximations. For the coupled nonlinear Hemholtz system we establish a general existence theory of the solutions for all frequencies and characterize their low frequency behaviour.

Keywords:

• Ferroelectric
• Nonlinear Maxwell equations
• Nonlinear Helmholtz equations
• Low frequency behaviour
• 1991 AMS Subject Classifications:
• 35L45
• 35L10
• 35K05
• 73R05
• 46E35