PDF(469 KB)
PDF(469 KB)
PDF(469 KB)
关于电磁场方程组解的W1,p正则性研究
On W1,p Regularity of A System Arising from Electromagnetic Fields
本文研究了R3中有界区域Ω上的电磁场方程组弱解的W1，p估计.该方程组来自于磁场所满足的稳态麦克斯韦方程组.在假定系数矩阵的逆属于VMO空间的条件下，利用R3中向量场的旋度和散度的性质，将该方程组转化为标量椭圆型方程组，从而根据椭圆型方程组的正则性理论，得到解的W1，p估计，其中1 < p < ∞.
We establish the fundamental W1,p estimate for the weak solution of a system in a bounded domain Ω in R3. The system is related to the steady-state of Maxwell's equations for the magnetic field. The inverse of the principle coefficient matrix is assumed to be in the VMO space. We transform the system to scalar elliptic equations by using the properties of curl and divergence of vector fields in R3. By the regularity theory of elliptic equations, we get the W1,p estimate for 1 < p < ∞.
W1 / p正则性 / 麦克斯韦方程 / VMO系数 {{custom_keyword}} /
W1,p regularity / Maxwell's equation / VMO coefficients {{custom_keyword}} /
[1] Alberti G. S., Yves C., Elliptic regularity theory applied to time harmonic anisotropic Maxwell's equations with less than Lipschitz complex coeffinicents, SIAM J. Math. Anal., 2014, 46(1):998-1016.
[2] Auscher P., Qafsaoui M., Observations on W1,p estimates for divergence elliptic equations with VMO coefficients, Bell. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2002, 5(8):487-509.
[3] Galdi G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I, Linearized Steady Problems, Springer-Verlag, New York, 1994.
[4] Giaquinta M., Hong M. C., Partial regularity of minimizers of a functional involving forms and maps, Nonlinear Differ. Equ. Appl., 2004, 11:469-490.
[5] Gilbarg D., Trudinger N., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
[6] Girault V., Ravirart P., Finite Element Methods for Navier-Stokes Equation, Springer-Verlag, Berlin, 1985.
[7] Kyungkeun K., Seick K., Elliptic systems with measurable coefficients of the type of Lamé system in the three dimensions, J. Differential Equations, 2011, 251:2466-2493.
[8] Kyungkeun K., Seick K., On the Hölder continuity of solutions of a certain system related to Maxwell's equations, SIAM J. Math. Anal., 2002, 34(1):87-100.
[9] Ladyzhenskaya O. A., Solonnikov V. A., Ural'ceva N. N., Linear and Quasilinear Elliptic Equations of Parabolic Type, Academic Press, New York, 1968.
[10] Landau L. D., Lifshitz E. M., Electrodynamics of Continuous Media, Pergamon Press, New York, 1960.
[11] Leis R., Initial-Boundary Value Problems in Mathematical Physics, B. G. Teubner, Stuttgart, 1986.
[12] Nguyen T., Wang J. N., Quantitative uniqueness estimate for the Maxwell system with Lipschitz anisotropic media, Proc. Amer. Math. Soc., 2012, 140:595-605.
[13] Sarason D., Functions of vanishing mean oscillation. Trans. Amer. Math. Soc., 1975, 207:391-405.
[14] Shen Z. W., Song L., On Lp estimates in homogenization of elliptic equations of Maxwell's type, Adv. in Math., 2014, 252:7-21.
[15] Yin H. M., Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields, Commun. Pure Appl. Anal., 2002, 1:127-134.
[16] Yin H. M., Regularity of solutions to Maxwell's system in quasi-stationary electromagnetic fields and applications, Comm. Partial Differential Equation, 1997, 22:1029-1053.
国家自然科学基金资助项目（11671316）
/
| 〈 |
|
〉 |
