March  2002, 1(1): 127-134. doi: 10.3934/cpaa.2002.1.127

Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields

1. 

Department of Mathematics, Washington State University, Pullman, WA 99164, United States

Received  February 2001 Revised  June 2001 Published  December 2001

In this paper we prove a fundamental estimate for the weak solution of a degenerate elliptic system: $\nabla\times [\rho(x)\nabla\times H]=F$, $\nabla\cdot H=0$ in a bounded domain in $R^3$, where $\rho(x)$ is only assumed to be in $L^{\infty}$ 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 $\rho(x)$ 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.
Citation: H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure & Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127
[1]

B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405

[2]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547

[3]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[4]

S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590

[5]

Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations & Control Theory, 2019, 8 (2) : 343-357. doi: 10.3934/eect.2019018

[6]

Luca Capogna. Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications. Electronic Research Announcements, 1996, 2: 60-68.

[7]

Giovanni Alessandrini, Maarten V. de Hoop, Romina Gaburro, Eva Sincich. EIT in a layered anisotropic medium. Inverse Problems & Imaging, 2018, 12 (3) : 667-676. doi: 10.3934/ipi.2018028

[8]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431

[9]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257

[10]

Panagiota Daskalopoulos, Eunjai Rhee. Free-boundary regularity for generalized porous medium equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 481-494. doi: 10.3934/cpaa.2003.2.481

[11]

Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1401-1407. doi: 10.3934/dcdss.2013.6.1401

[12]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299

[13]

Ilaria Fragalà, Filippo Gazzola, Gary Lieberman. Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains. Conference Publications, 2005, 2005 (Special) : 280-286. doi: 10.3934/proc.2005.2005.280

[14]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117

[15]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159

[16]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181

[17]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607

[18]

Serge Nicaise, Fredi Tröltzsch. Optimal control of some quasilinear Maxwell equations of parabolic type. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1375-1391. doi: 10.3934/dcdss.2017073

[19]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229

[20]

Yumei Liao, Wei Wei, Xianbing Luo. Existence of solution of a microwave heating model and associated optimal frequency control problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019045

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]