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Mixed boundary-value problems for Maxwell's equations

Author: Marius Mitrea
Journal: Trans. Amer. Math. Soc. 362 (2010), 117-143
MSC (2000): Primary 35J55, 78A30, 42B20; Secondary 35F15, 35C15, 78M15
Published electronically: August 13, 2009
MathSciNet review: 2550146
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Abstract: We study the Maxwell system with mixed boundary conditions in a Lipschitz domain $ \Omega$ in $ \mathbb{R}^3$. It is assumed that two disjoint, relatively open subsets $ \Sigma^e$, $ \Sigma^h$ of $ \partial\Omega$ such that $ \overline{\Sigma^e}\cap\overline{\Sigma^h}=\partial\Sigma^e=\partial\Sigma^h$ have been fixed, and one prescribes the tangential components of the electric and magnetic fields on $ \Sigma^e$ and $ \Sigma^h$, respectively. Under suitable geometric assumptions on $ \partial\Omega$, $ \Sigma^e$ and $ \Sigma^h$, we prove that this boundary value problem is well-posed when $ L^p$-estimates for the nontangential maximal function are sought, with $ p$ near $ 2$. A higher-dimensional version of this result is established as well, in the language of differential forms. This extends earlier work by R. Brown and by the author and collaborators.

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  • 1. A. Alonso Rodríguez and M. Raffetto, Unique solvability for electromagnetic boundary value problems in the presence of partly lossy inhomogeneous anisotropic media and mixed boundary conditions, Math. Models Methods Appl. Sci., 13 (2003), no. 4, 597-611. MR 1976304 (2004c:35396)
  • 2. G.A. Baker and J. Dodziuk, Stability of spectra of Hodge-de Rham Laplacians, Math. Z., 224 (1997), no. 3, 327-345. MR 1439194 (98h:58191)
  • 3. R. Brown, The mixed problem for Laplace's equation in a class of Lipschitz domains, Comm. Partial Differential Equations, 19 (1994), no. 7-8, 1217-1233. MR 1284808 (95i:35059)
  • 4. F. Cakoni and D. Colton, Mixed boundary value problems in inverse electromagnetic scattering, Proceedings of the 6th International Workshop on Mathematical Methods in Scattering Theory and Biomedical Engineering, Tsepelovo, 2003.
  • 5. F. Cakoni, D. Colton and P. Monk, Electromagnetic inverse scattering problem for partially coated Lipschitz Domains, to appear in Proc. Royal Society of Edinburgh. MR 2079799 (2005g:78016)
  • 6. R.R. Coifman, A. McIntosh and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $ L^2$ pour les courbes lipschitziennes, Ann. of Math. (2), 116 (1982), no. 2, 361-387. MR 672839 (84m:42027)
  • 7. B. Dahlberg and C. Kenig, Hardy spaces and the $ L^p$-Neumann problem for Laplace's equation in a Lipschitz domain, Ann. of Math., 125 (1987), 437-465. MR 890159 (88d:35044)
  • 8. E.B. Fabes, M. Jodeit, Jr. and N.M. Rivière, Potential techniques for boundary value problems on $ C^1$-domains, Acta Math., 141 (1978), no. 3-4, 165-186. MR 501367 (80b:31006)
  • 9. P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci., 7 (1997), no. 7, 957-991. MR 1479578 (98h:78006)
  • 10. H.G. Garnir, Les problémes aux limites de la physique mathématique, Basel, Birkhäuser Verlag, 1958. MR 0104054 (21:2816)
  • 11. S. Hofmann, On singular integrals of Calderón-type in $ \mathbb{R}^n$, and $ BMO$, Rev. Mat. Iberoamericana, 10 (1994), no. 3, 467-505. MR 1308701 (96d:42019)
  • 12. B. Jawerth and M. Mitrea, Higher-dimensional electromagnetic scattering theory on $ C^1$ and Lipschitz domains, Amer. J. Math., 117 (1995), no. 4, 929-963. MR 1342836 (96h:35143)
  • 13. F. Jochmann, Existence of weak solutions of the drift diffusion model coupled with Maxwell's equations, J. Math. Anal. Appl., 204 (1996), no. 3, 655-676. MR 1422765 (98e:35030)
  • 14. F. Jochmann, Uniqueness and regularity for the two-dimensional drift-diffusion model for semiconductors coupled with Maxwell's equations, J. Differential Equations, 147 (1998), no. 2, 242-270. MR 1634016 (99g:35125)
  • 15. F. Jochmann, Regularity of weak solutions of Maxwell's equations with mixed boundary-conditions, Math. Methods Appl. Sci., 22 (1999), no. 14, 1255-1274. MR 1710708 (2000g:78008)
  • 16. C.E. Kenig, Recent progress on boundary value problems on Lipschitz domains, Pseudodifferential operators and applications (Notre Dame, Ind., 1984), pp. 175-205, Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI, 1985. MR 812291 (87e:35029)
  • 17. D. Mitrea and M. Mitrea, Finite energy solutions of Maxwell's equations and constructive Hodge decompositions on nonsmooth Riemannian manifolds, J. Funct. Anal., 190 (2002), no. 2, 339-417. MR 1899489 (2003m:58002)
  • 18. D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on nonsmooth domains in $ \mathbb{R}^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl., 3 (1997), no. 2, 131-192. MR 1438894 (99e:31009)
  • 19. D. Mitrea, M. Mitrea and M. Taylor, Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds, Mem. Amer. Math. Soc., Vol. 150 No. 713, 2001. MR 1809655 (2002g:58026)
  • 20. M. Mitrea, The method of layer potentials in electromagnetic scattering theory on nonsmooth domains, Duke Math. J., 77 (1995), no. 1, 111-133. MR 1317629 (96b:78035)
  • 21. M. Mitrea and M. Muether, An integration by parts formula in submanifolds of positive codimension, Math. Methods Appl. Sci., Vol. 27, No. 4 (2004), 1711-1723. MR 2089158 (2005i:58008)
  • 22. J.D. Sykes and R.M. Brown, The mixed boundary problem in $ L^p$ and Hardy spaces for Laplace's equation on a Lipschitz domain, Harmonic analysis and boundary value problems (Fayetteville, AR, 2000), pp. 1-18, Contemp. Math., 277, Amer. Math. Soc., Providence, RI, 2001. MR 1840423 (2002g:35058)
  • 23. M. Taylor, Partial Differential Equations, Vol. I-III, Applied Mathematical Sciences, 116, Springer-Verlag, New York, 1996. MR 1395149 (98b:35003)

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Additional Information

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri at Columbia, Columbia, Missouri 65211

Keywords: Maxwell's equations, Lipschitz domains, mixed boundary conditions, Rellich estimates.
Received by editor(s): June 21, 2005
Received by editor(s) in revised form: April 12, 2007
Published electronically: August 13, 2009
Additional Notes: The author was supported in part by the NSF and the University of Missouri Office of Research
Article copyright: © Copyright 2009 American Mathematical Society

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