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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

Matthias Eller

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

Received  May 2018 Revised  September 2018 Published  June 2019 Early access  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.

Keywords: Maxwell equations, stability, initial-boundary value problem.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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.   Google Scholar
[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

show all references

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.   Google Scholar
[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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