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June  2019, 8(2): 343-357. doi: 10.3934/eect.2019018

## Stability of the anisotropic Maxwell equations with a conductivity term

 Department of Mathematics, Georgetown University, Washington, DC 20057, USA

Received  May 2018 Revised  September 2018 Published  March 2019

The dynamic Maxwell equations with a conductivity term are considered. Conditions for the exponential and strong stability of an initial-boundary value problem are given. The permeability and the permittivity are assumed to be $3\times 3$ symmetric, positive definite tensors. A result concerning solutions of higher regularity is obtained along the way.

Citation: 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
##### References:
 [1] M. Belishev and A. Glasman, Boundary control of the Maxwell dynamical system: Lack of controllability by topological reasons, in Mathematical and Numerical Aspects of Wave Propagation—WAVES 2003, Springer, Berlin, 2003,177-182. Google Scholar [2] J. Cagnol and M. Eller, Boundary regularity for Maxwell's equations with applications to shape optimization, J. Differential Equations, 250 (2011), 1114-1136. doi: 10.1016/j.jde.2010.08.004. Google Scholar [3] M. Cessenat, Mathematical Methods in Electromagnetism, vol. 41 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co. Inc., River Edge, NJ, 1996, Linear theory and applications. doi: 10.1142/2938. Google Scholar [4] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990, Spectral theory and applications, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson. Google Scholar [5] W. Desch, R. Grimmer and W. Schappacher, Some considerations for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234. doi: 10.1016/0022-247X(84)90044-1. Google Scholar [6] M. Eller and D. Toundykov, A global holmgren theorem for multidimensional hyperbolic partial differential equations, Applicable Analysis, 91 (2012), 69-90. doi: 10.1080/00036811.2010.538685. Google Scholar [7] M. M. Eller, Continuous observability for the anisotropic Maxwell system, Appl. Math. Optim., 55 (2007), 185-201. doi: 10.1007/s00245-006-0886-x. Google Scholar [8] M. M. Eller and M. Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system, J. Math. Pures Appl. (9), 86 (2006), 449-462. doi: 10.1016/j.matpur.2006.10.004. Google Scholar [9] N. Iwasaki, Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains, Publ. Res. Inst. Math. Sci., 5 (1969), 193-218. doi: 10.2977/prims/1195194630. Google Scholar [10] P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. [11] R. Leis, über die eindeutige Fortsetzbarkeit der Lösungen der Maxwellschen Gleichungen in anisotropen inhomogenen Medien, Bul. Inst. Politehn. Iașsi (N.S.), 14 (1968), 119-124. Google Scholar [12] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504. Google Scholar [13] C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14 (1961), 561-568. doi: 10.1002/cpa.3160140327. Google Scholar [14] S. Nicaise and C. Pignotti, Internal stabilization of Maxwell's equations in heterogeneous media, Abstr. Appl. Anal., 2005,791-811. doi: 10.1155/AAA.2005.791. Google Scholar [15] T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary, Hokkaido Math. J., 10 (1981), 93-123. doi: 10.14492/hokmj/1381758116. Google Scholar [16] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [17] K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137. doi: 10.1051/cocv:2000103. Google Scholar [18] R. Picard and W. Zajaczkowski, Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, Mathematical Methods in the Applied Sciences, 18 (1995), 169-199. doi: 10.1002/mma.1670180302. Google Scholar [19] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, vol. 133 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/133. Google Scholar [20] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86, URL https://doi.org/10.1512/iumj.1974.24.24004. doi: 10.1512/iumj.1975.24.24004. Google Scholar [21] J. B. Rauch and F. J. Massey Ⅲ, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861. Google Scholar [22] V. A. Solonnikov, Overdetermined elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 199 (1971), 279-281. Google Scholar [23] M. Spitz, Local Wellposedness of Nonlinear Maxwell Equations, Karlsruhe Institute of Technology, https://doi.org/10.5445/IR/1000078030, 2017, Ph.D Thesis.Google Scholar

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##### References:
 [1] M. Belishev and A. Glasman, Boundary control of the Maxwell dynamical system: Lack of controllability by topological reasons, in Mathematical and Numerical Aspects of Wave Propagation—WAVES 2003, Springer, Berlin, 2003,177-182. Google Scholar [2] J. Cagnol and M. Eller, Boundary regularity for Maxwell's equations with applications to shape optimization, J. Differential Equations, 250 (2011), 1114-1136. doi: 10.1016/j.jde.2010.08.004. Google Scholar [3] M. Cessenat, Mathematical Methods in Electromagnetism, vol. 41 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co. Inc., River Edge, NJ, 1996, Linear theory and applications. doi: 10.1142/2938. Google Scholar [4] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990, Spectral theory and applications, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson. Google Scholar [5] W. Desch, R. Grimmer and W. Schappacher, Some considerations for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234. doi: 10.1016/0022-247X(84)90044-1. Google Scholar [6] M. Eller and D. Toundykov, A global holmgren theorem for multidimensional hyperbolic partial differential equations, Applicable Analysis, 91 (2012), 69-90. doi: 10.1080/00036811.2010.538685. Google Scholar [7] M. M. Eller, Continuous observability for the anisotropic Maxwell system, Appl. Math. Optim., 55 (2007), 185-201. doi: 10.1007/s00245-006-0886-x. Google Scholar [8] M. M. Eller and M. Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system, J. Math. Pures Appl. (9), 86 (2006), 449-462. doi: 10.1016/j.matpur.2006.10.004. Google Scholar [9] N. Iwasaki, Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains, Publ. Res. Inst. Math. Sci., 5 (1969), 193-218. doi: 10.2977/prims/1195194630. Google Scholar [10] P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. [11] R. Leis, über die eindeutige Fortsetzbarkeit der Lösungen der Maxwellschen Gleichungen in anisotropen inhomogenen Medien, Bul. Inst. Politehn. Iașsi (N.S.), 14 (1968), 119-124. Google Scholar [12] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504. Google Scholar [13] C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14 (1961), 561-568. doi: 10.1002/cpa.3160140327. Google Scholar [14] S. Nicaise and C. Pignotti, Internal stabilization of Maxwell's equations in heterogeneous media, Abstr. Appl. Anal., 2005,791-811. doi: 10.1155/AAA.2005.791. Google Scholar [15] T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary, Hokkaido Math. J., 10 (1981), 93-123. doi: 10.14492/hokmj/1381758116. Google Scholar [16] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [17] K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137. doi: 10.1051/cocv:2000103. Google Scholar [18] R. Picard and W. Zajaczkowski, Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, Mathematical Methods in the Applied Sciences, 18 (1995), 169-199. doi: 10.1002/mma.1670180302. Google Scholar [19] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, vol. 133 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/133. Google Scholar [20] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86, URL https://doi.org/10.1512/iumj.1974.24.24004. doi: 10.1512/iumj.1975.24.24004. Google Scholar [21] J. B. Rauch and F. J. Massey Ⅲ, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861. Google Scholar [22] V. A. Solonnikov, Overdetermined elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 199 (1971), 279-281. Google Scholar [23] M. Spitz, Local Wellposedness of Nonlinear Maxwell Equations, Karlsruhe Institute of Technology, https://doi.org/10.5445/IR/1000078030, 2017, Ph.D Thesis.Google Scholar
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