Article
MSC:
35A15,
35D30,
47H05,
65N30,
78M10,
78M30 | MR 4444786 | Zbl 07584079 | DOI: 10.21136/AM.2021.0365-20
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Keywords:
well-posedness; uniform monotonicity; S-property; $p$-curl systems
well-posedness; uniform monotonicity; S-property; $p$-curl systems
Summary:
In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic $p$-curl systems.
References:
[1] Antontsev, S., Miranda, F., Santos, L.: A class of electromagnetic $p$-curl systems: Blow-up and finite time extinction. Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 3916-3929. DOI 10.1016/j.na.2012.02.011 | MR 2914580 | Zbl 1246.35102
[2] Antontsev, S., Miranda, F., Santos, L.: Blow-up and finite time extinction for $p(x,t)$-curl systems arising in electromagnetism. J. Math. Anal. Appl. 440 (2016), 300-322 \99999DOI99999 10.1016/j.jmaa.2016.03.045 . DOI 10.1016/j.jmaa.2016.03.045 | MR 3479601 | Zbl 1339.35060
[3] Aramaki, J.: $L^p$ theory for the div-curl system. Int. J. Math. Anal., Ruse 8 (2014), 259-271 \99999DOI99999 10.12988/ijma.2014.4112 . MR 3188605
[4] Bahrouni, A., Repovš, D.: Existence and nonexistence of solutions for $p(x)$-curl systems arising in electromagnetism. Complex Var. Elliptic Equ. 63 (2018), 292-301 \99999DOI99999 10.1080/17476933.2017.1304390 . MR 3764762 | Zbl 1423.35124
[5] Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, Berlin (2010),\99999DOI99999 10.1007/978-1-4419-5542-5 . MR 2582280 | Zbl 1197.35002
[6] Cimrák, I., Keer, R. Van: Level set method for the inverse elliptic problem in nonlinear electromagnetism. J. Comput. Phys. 229 (2010), 9269-9283 \99999DOI99999 10.1016/j.jcp.2010.08.038 . MR 2733153 | Zbl 1207.78045
[7] Dunford, N., Schwartz, J. T.: Linear Operators. I. General Theory. Pure and Applied Mathematics 7. Interscience Publishers, New York (1958),\99999MR99999 0117523 . MR 0117523 | Zbl 0084.10402
[8] u, J. Franc\accent23: Monotone operators: A survey directed to applications to differential equations. Appl. Math. 35 (1990), 257-301 \99999DOI99999 10.21136/AM.1990.104411 . MR 1065003 | Zbl 0724.47025
[9] Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Mathematische Lehrbücher und Monographien. II. Abteilung. Band 38. Akademie, Berlin (1974), German \99999MR99999 0636412 . MR 0636412 | Zbl 0289.47029
[10] Gerbeau, J.-F., Bris, C. Le, Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2006),\99999DOI99999 10.1093/acprof:oso/9780198566656.001.0001 . MR 2289481 | Zbl 1107.76001
[11] Janíková, E., Slodička, M.: Fully discrete linear approximation scheme for electric field diffusion in type-II superconductors. J. Comput. Appl. Math. 234 (2010), 2054-2061 \99999DOI99999 10.1016/j.cam.2009.08.063 . MR 2652398 | Zbl 1195.82105
[12] László, S. C.: The Theory of Monotone Operators with Applications. Babes-Bolyai University, Budapest (2011) .
[13] Xiang, M., Wang, F., Zhang, B.: Existence and multiplicity for $p(x)$-curl systems arising in electromagnetism. J. Math. Anal. Appl. 448 (2017), 1600-1617 \99999DOI99999 10.1016/j.jmaa.2016.11.086 . MR 3582298 | Zbl 1358.35181
[14] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators. Springer, New York (1990). DOI 10.1007/978-1-4612-0981-2 | MR 1033498 | Zbl 0684.47029
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