No abstract

No abstract

In this paper we prove a fundamental estimate for the weak solution of a degenerate elliptic system: ∇ × ρ(x)∇ × H = F, ∇ • H = 0 in a bounded domain in R 3 , where ρ(x) is only assumed to be in L ∞ with a positive lower bound. This system is the steady-state of Maxwell's system for the evolution of a magnetic field H under the influence of an external force F, where ρ(x) represents the resistivity of the conductive material. By using Campanato type of techniques, we show that the weak solution to the system is Hölder continuous, which is optimal under the assumption. This result solves the regularity problem for the system under the minimum assumption on the coefficient. Some applications arising in inductive heating are presented.

Abstract. The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operators with Navier boundary value conditions. The technical approach is mainly based on a three critical points theorem due to Ricceri [11].

In this paper, we study the existence of three solutions to a Neumann problem with nonstandard growth conditions. The technical approach is mainly based on three critical points theorem due to Ricceri. Copyright © 2011 John Wiley & Sons, Ltd.