**MSC2010** 35Q61

### Modified gauge conditions for Maxwell equations in quasi-stationary magnetic approximation

#### A. B. Kalinin^{1}, A. A. Tyukhtina^{2}, O. A. Izosimova^{3}

Annotation | The initial-boundary value problem for the Maxwell equations in the quasi-stationary magnetic approximation with magnetic boundary conditions is studied. The case of heterogeneous media containing conductive and non-conductive inclusions is considered. The problem is formulated in terms of vector magnetic and scalar electric potentials. The special gauge conditions (modified Coulomb and Lorenz gauges) that allow to formulate problems of independent determination of vector magnetic potential are proposed. The correctness of the problem statement under general conditions on coefficients is proved. The possibility of Lions’ theorem application is proved using estimates for scalar products of vector fields. The relation between solutions of the problems with different gauges is studied. The correspondence between the problem formulation for magnetic vector potential and the initial problem formulation for the vector magnetic potential and scalar electric potential is considered. |
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Keywords | Maxwell equations, quasi-stationary magnetic approximation, heterogeneous media, vector potential, gauge conditions |

^{1}**Aleksey V. Kalinin**, Associate Professor of the Chair of Mathematical Phisics and Optimal Control, Institute of Informational Technology, Mathematics and Mechanics, N.I. Lobachevsky State University (23 Gagarin Av., Nizny Novgorod 603950, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-8558-0451, avk@mm.unn.ru^{2}**Alla A. Tyukhtina**, Associate Professor of the Chair of Mathematical Modelling of Economical Processes, Institute of Economics and Entrepreneurship, N.I. Lobachevsky State University (23 Gagarin Av., Nizny Novgorod 603950, Russia), Ph.D. (Physics and Mathematics), ORCID:http://orcid.org/0000-0001-5723-9256, kalinmm@yandex.ru^{3}**Olga A. Izosimova**, Graduate student, Institute of Informational Technology, Mathematics and Mechanics, N.I. Lobachevsky State University (23 Gagarin Av., Nizny Novgorod 603950, Russia), ORCID: http://orcid.org/0000-0003-3865-8281, izosimova93@yandex.ru

**Citation**: A. B. Kalinin, A. A. Tyukhtina, O. A. Izosimova, "[Modified gauge conditions for Maxwell equations in quasi-stationary magnetic approximation]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:4 (2017) 55–67 (In Russian)

**DOI** 10.15507/2079-6900.19.201704.55-67