小测度缺陷下Maxwell方程解的唯一性
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摘要
在基于已知背景区域内Maxwell方程解的唯一性基础上,研究了区域内存在缺陷时解的唯一性.通过讨论不同边值对应不同特征值的特点获得了当缺陷在测度足够小时解的唯一性依旧成立的结论,并且将该结论延伸至相对一般结构的小测度缺陷的情况.
Under the condition that the uniqueness of the solutions of Maxwell's equations in the background region is known, this paper studies the uniqueness of the solutions in the region in the presence of defects. By studying the eigenvalue problems for different boundary conditions, we obtain the uniqueness in the presence of defects with small measurement, and then we extent the result to general small measurement defects.
Under the condition that the uniqueness of the solutions of Maxwell's equations in the background region is known, this paper studies the uniqueness of the solutions in the region in the presence of defects. By studying the eigenvalue problems for different boundary conditions, we obtain the uniqueness in the presence of defects with small measurement, and then we extent the result to general small measurement defects.
引文
[1] BALL J M,CAPDEBOSCQ Y,TSERING-XIAO B.On Uniqueness for Time Harmonic Anisotropic Maxwell’s Equations with Piecewise Regular Coefficients [J].Mathematical Models and Methods in Applied Sciences,2012,22(11):364-375.
[2] DEGENNES G P,PROST J.The Physics of Liquid Crystals [M] //International Series of Monographs on Physics.Oxford:Oxford Science Publications,1995.
[3] KLEMAN M.Defects in Liquid Crystals [J].Reports on Progress in Physics,1989,52(5):555-654.
[4] KLEMAN M,LAVRENTOVICH O D.Topological Point Defects in Nematic Liquid Crystals [J].Philosophical Magazine,2006,86(25-26):4117-4137.
[5] MAJUMDAR A.Equilibrium Order Parameters of Nematic Liquid Crystals in the Landau-De Gennes Theory [J].European Journal of Applied Mathematics,2010,21(2):181-203.
[6] 小巴桑次仁.RELLICH引理的新证明 [J].数学物理学报(A辑),2014,34(6):1435-1439.
[7] 小巴桑次仁.各项同性麦克斯韦方程在系数为弱正则时解的唯一性证明 [J].西南师范大学学报(自然科学版),2013,38(10):1-4.
[8] 庞礼军.麦克斯韦方程组与其它普遍规律的等价性 [J].贵州师范大学学报(自然科学版),2000(2):44-46.
[9] AMMARI H,KANG H.High-Order Terms in the Asymptotic Expansions of the Steady-State Voltage Potentials in the Presence of Conductivity Inhomogeneities of Small Diameter [J].SIAM Journal on Mathematical Analysis,2002,34(5):1152-1166.
[10] AMMARI H,VOGELIUS M S,VOLKOV D.Asymptotic Formulas for Perturbations in the Electromagnetic Fields Due to the Presence of Inhomogeneities of Small Diameter II,the Full Maxwell Equations [J].Journal De Mathématiques Pures Et Appliquées,2001,80(8):769-814.
[11] GRIESMAIER R.A General Perturbation Formula for Electromagnetic Fields in Presence of Low Volume Scatterers [J].ESAIM:Mathematical Modelling and Numerical Analysis,2011,45(6):1193-1218.
[12] CAPDEBOSCQ Y,VOGELIUS M S.A General Representation Formula for Boundary Voltage Perturbations Caused by Internal Conductivity Inhomogeneities of Low Volume Fraction [J].ESAIM:Mathematical Modelling and Numerical Analysis,2003,37(1):159-173.
[13] AMMARI H,KHELIFI A.Electromagnetic Scattering by Small Dielectric Inhomogeneities [J].Journal De Mathématiques Pures Et Appliquées,2003,82(7):749-842.
[2] DEGENNES G P,PROST J.The Physics of Liquid Crystals [M] //International Series of Monographs on Physics.Oxford:Oxford Science Publications,1995.
[3] KLEMAN M.Defects in Liquid Crystals [J].Reports on Progress in Physics,1989,52(5):555-654.
[4] KLEMAN M,LAVRENTOVICH O D.Topological Point Defects in Nematic Liquid Crystals [J].Philosophical Magazine,2006,86(25-26):4117-4137.
[5] MAJUMDAR A.Equilibrium Order Parameters of Nematic Liquid Crystals in the Landau-De Gennes Theory [J].European Journal of Applied Mathematics,2010,21(2):181-203.
[6] 小巴桑次仁.RELLICH引理的新证明 [J].数学物理学报(A辑),2014,34(6):1435-1439.
[7] 小巴桑次仁.各项同性麦克斯韦方程在系数为弱正则时解的唯一性证明 [J].西南师范大学学报(自然科学版),2013,38(10):1-4.
[8] 庞礼军.麦克斯韦方程组与其它普遍规律的等价性 [J].贵州师范大学学报(自然科学版),2000(2):44-46.
[9] AMMARI H,KANG H.High-Order Terms in the Asymptotic Expansions of the Steady-State Voltage Potentials in the Presence of Conductivity Inhomogeneities of Small Diameter [J].SIAM Journal on Mathematical Analysis,2002,34(5):1152-1166.
[10] AMMARI H,VOGELIUS M S,VOLKOV D.Asymptotic Formulas for Perturbations in the Electromagnetic Fields Due to the Presence of Inhomogeneities of Small Diameter II,the Full Maxwell Equations [J].Journal De Mathématiques Pures Et Appliquées,2001,80(8):769-814.
[11] GRIESMAIER R.A General Perturbation Formula for Electromagnetic Fields in Presence of Low Volume Scatterers [J].ESAIM:Mathematical Modelling and Numerical Analysis,2011,45(6):1193-1218.
[12] CAPDEBOSCQ Y,VOGELIUS M S.A General Representation Formula for Boundary Voltage Perturbations Caused by Internal Conductivity Inhomogeneities of Low Volume Fraction [J].ESAIM:Mathematical Modelling and Numerical Analysis,2003,37(1):159-173.
[13] AMMARI H,KHELIFI A.Electromagnetic Scattering by Small Dielectric Inhomogeneities [J].Journal De Mathématiques Pures Et Appliquées,2003,82(7):749-842.