Differential and Integral Equations

Regularity of weak solution to a $p$-curl-system

Hong-Ming Yin

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In this note we study the regularity of weak solutions to a nonlinear steady-state Maxwell's equation in conductive media: $\nabla\times \big[|{\nabla \times }{{\bf H}}|^{p-2} \nabla \times {{\bf H}} \big]={ {\bf F} }(x), p>1, $ where ${{\bf H}}(x)$ represents the magnetic field while ${ {\bf F} }(x)$ is the internal magnetic current. It is shown that the weak solution to the above system is of class $C^{1+\alpha}$, which is optimal. The basic idea is to introduce a suitable potential and then to transform the system into a $p-$Laplacian equation subject to a Neumann type of boundary condition. The desired regularity is established by using the known theory for the scalar p-Laplacian equation.

Article information

Differential Integral Equations, Volume 19, Number 4 (2006), 361-368.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q60: PDEs in connection with optics and electromagnetic theory
Secondary: 35D10 35J50: Variational methods for elliptic systems 49N60: Regularity of solutions 78A25: Electromagnetic theory, general


Yin, Hong-Ming. Regularity of weak solution to a $p$-curl-system. Differential Integral Equations 19 (2006), no. 4, 361--368. https://projecteuclid.org/euclid.die/1356050504

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