Variational problem involving operator curl associated with p-curl system

We shall study the problem of minimizing a functional involving curl of vector fields in a three-dimensional,bounded multiconnected domain with the prescribed tangent component of a given vector field on the boundary. It willbe seen that the minimizers are weak solutions of the p-curl type system. We shall prove the existence and the estimateof minimizers of a more general functional that contains the $L^p$ norm of the curl of vector fields. We shall also give thecontinuity with respect to the given data.

PDF

___

[1] Amrouche C, Seloula NH. L p -theory for vector potentials and Sobolev’s inequalities for vector fields. C R Acad Sci Paris Ser 1 2011; 349: 529-534.
[2] Amrouche C, Seloula NH. L p -theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math Models Methods Appl Sci 2013; 23: 37092.
[3] Aramaki J. Minimizing divergence of vector fields in a multi-connected domain. Far East J Math Sci 2010; 38: 65-84.
[4] Aramaki J. Regularity of weak solutions for degenerate quasilinear elliptic equations involving curl. J Math Anal Appl 2015; 425: 872-892.
[5] Bates P, Pan XB. Nucleation of instability of the Meissner state of 3 -dimensional superconductors. Commun Math Phys 2007; 276: 571-609.
[6] Dautray R, Lions JL. Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3. New York, NY, USA: Springer-Verlag, 1990.
[7] DiBenedetto E. Degenerate Parabolic Equations. Berlin, Germany: Springer-Science+Business Media, 1993.
[8] Evans LC. Weak Convergence Methods for Nonlinear Partial Differential Equations. Regional Conference Series in Mathematics Number 74. Providence, RI, USA: AMS, 1988.
[9] Fujita H, Kuroda N, Ito S. Functional Analysis. Tokyo, Japan: Iwanami Shoten, 1990 (in Japanese).
[10] Giraut V, Raviart PA. Finite Element Methods for Navier-Stokes Equations. Berlin, Germany: Springer-Verlag, 1986.
[11] Miranda F, Rodrigues JF, Santos L. On a p-curl system arising in electromagnetism. Discrete Cont Dynamical System Ser S 2012; 5: 605-629.
[12] Pan XB. Minimizing curl in a multiconnected domain. J Math Phys 2009; 50: 1-10.
[13] Pan XB, Qi YW. Asymptotics of minimizers of variational problem involving curl functional. J Math Phys 2000; 41: 5033.
[14] Sobolev S. Applications of Functional Analysis in Mathematical Physics. Translations of Mathematical Monographs 7. Providence, RI, USA: American Mathematical Society, 1963.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv