Geodesic
Elenbaas Problem of Electric Arc Discharge
Matematičeskie zametki, Tome 103 (2018) no. 1, pp. 92-100.

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The Elenbaas problem of electric discharge origination is considered. The mathematical model is an elliptic boundary-value problem with a parameter and discontinuous nonlinearity. The nontrivial solutions of the problem determine the free boundaries separating different phase states. A survey of results obtained for this problem is given. The greatest lower bound $\lambda_{\min}$ of the values of the parameter $\lambda$ for which the electric discharge is possible is obtained. The fact that the discharge domain appears for any $\lambda \ge \lambda_{\min}$ is proved. The range of the parameter values for which the boundary of the discharge domain is of two-dimensional Lebesgue measure zero is determined. An unsolved problem is formulated.
Keywords: Elenbaas problem, electric arc, free boundary, discontinuous nonlinearity. 
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V. N. Pavlenko; D. K. Potapov. Elenbaas Problem of Electric Arc Discharge. Matematičeskie zametki, Tome 103 (2018) no. 1, pp. 92-100. http://geodesic.mathdoc.fr/item/MZM_2018_103_1_a7/

[1] V. Finkelnburg, G. Mekker, Elektricheskie dugi i termicheskaya plazma, IL, M., 1961

[2] W. Elenbaas, “The high pressure mercury vapour discharge”, High Pressure Mercury Vapour Lamps and Their Applications, Palgrave, London, 1965, 5–51 | DOI

[3] D. K. Potapov, “Nepreryvnye approksimatsii zadachi Goldshtika”, Matem. zametki, 87:2 (2010), 262–266 | DOI | MR | Zbl

[4] D. K. Potapov, “O resheniyakh zadachi Goldshtika”, Sib. zhurn. vychisl. matem., 15:4 (2012), 409–415 | Zbl

[5] D. K. Potapov, V. V. Yevstafyeva, “Lavrent'ev problem for separated flows with an external perturbation”, Electron. J. Differential Equations, 2013, no. 255 | MR | Zbl

[6] D. K. Potapov, “Ob odnoi zadache elektrofiziki s razryvnoi nelineinostyu”, Differents. uravneniya, 50:3 (2014), 421–424 | DOI | MR | Zbl

[7] G. Cimatti, “A nonlinear elliptic eigenvalue problem for the Elenbaas equation”, Boll. Un. Mat. Ital. B (5), 16:2 (1979), 555–565 | MR | Zbl

[8] K.- C. Chang, “Free boundary problems and the set-valued mappings”, J. Differential Equations, 49:1 (1983), 1–28 | DOI | MR | Zbl

[9] W. Allegretto, P. Nistri, “Elliptic equations with discontinuous nonlinearities”, Topol. Methods Nonlinear Anal., 2:2 (1993), 233–251 | DOI | MR | Zbl

[10] A. Ambrosetti, M. Calahorrano, F. Dobarro, “Global branching for discontinuous problems”, Comment. Math. Univ. Carolin., 31:2 (1990), 213–222 | MR | Zbl

[11] G. Bonanno, “Some remarks on a three critical points theorem”, Nonlinear Anal., 54:4 (2003), 651–665 | DOI | MR | Zbl

[12] D. K. Potapov, “Ob odnoi otsenke sverkhu velichiny bifurkatsionnogo parametra v zadachakh na sobstvennye znacheniya dlya uravnenii ellipticheskogo tipa s razryvnymi nelineinostyami”, Differents. uravneniya, 44:5 (2008), 715–716 | MR | Zbl

[13] D. K. Potapov, “O strukture mnozhestva sobstvennykh znachenii dlya uravnenii ellipticheskogo tipa vysokogo poryadka s razryvnymi nelineinostyami”, Differents. uravneniya, 46:1 (2010), 150–152 | MR | Zbl

[14] D. K. Potapov, “O “razdelyayuschem” mnozhestve dlya uravnenii ellipticheskogo tipa vysokogo poryadka s razryvnymi nelineinostyami”, Differents. uravneniya, 46:3 (2010), 451–453 | MR | Zbl

[15] D. K. Potapov, “Bifurkatsionnye zadachi dlya uravnenii ellipticheskogo tipa s razryvnymi nelineinostyami”, Matem. zametki, 90:2 (2011), 280–284 | DOI | MR | Zbl

[16] D. Gilbarg, N. Trudinger, Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[17] I. Massabó, C. A. Stuart, “Elliptic eigenvalue problems with discontinuous nonlinearities”, J. Math. Anal. Appl., 66:2 (1978), 261–281 | DOI | MR | Zbl

[18] M. A. Krasnoselskii, A. V. Pokrovskii, Sistemy s gisterezisom, Nauka, M., 1983 | MR | Zbl

[19] V. N. Pavlenko, “Upravlenie singulyarnymi raspredelennymi sistemami parabolicheskogo tipa s razryvnymi nelineinostyami”, Ukr. matem. zhurn., 46:6 (1994), 729–736 | MR | Zbl

[20] V. N. Pavlenko, O. V. Ulyanova, “Metod verkhnikh i nizhnikh reshenii dlya uravnenii ellipticheskogo tipa s razryvnymi nelineinostyami”, Izv. vuzov. Matem., 1998, no. 11, 69–76 | MR | Zbl