Well-posedness of the Tricomi Problem for the Multidimensional Lavrent'ev-Bitsadze Equation
DOI:
https://doi.org/10.70474/05m52x96Keywords:
Tricomi problem, mixed domain, classical solution, multidimensional Lavrent'ev-Bitsadze equation, spherical functionsAbstract
Numerous applications in physics and engineering involve models with partial differential equations of mixed type. The theory of boundary-value problems for such equations in two dimensions has been well studied. However, the key problem of well-posedness of mixed problems for such equations in multidimensional bounded domains remains currently unsolved. This paper establishes a mixed domain in which the solution of the Tricomi problem for the multidimensional Lavrent'ev-Bitsadze equation has a unique classical solution.
Downloads
References
[1] Otway, T. H., 2010, Unique solutions to boundary value problems in the cold plasma model, SIAM Journal on Mathematical Analysis, vol. 42(6), pp. 3045-3053. DOI: https://doi.org/10.1137/090775786
[2] Bitsadze, A.V., Nekotorye klassy uravnenii v chastnykh proizvodnykh (Some Classes of Partial Differential Equations), Moscow: Nauka, 1981.
[3] Nakhushev, A.M., Zadachi so smeshcheniem dlya uravneniya v chastnykh proizvodnykh (Problems with Shift for Partial Differential Equations), Moscow: Nauka, 2006.
[4] Bitsadze, A.V., Uravneniya smeshannogo tipa (Mixed Type Equations), Moscow: Izd. Akad. Nauk SSSR, 1959.
[5] Bitsadze, A.V., On the problem of equations of mixed type in multidimensional domains, Dokl. Akad. Nauk SSSR, 1956, vol. 110, no. 6, pp. 901–902.
[6] Pul’kin, S.V., Singular Tricomi problem, in Tr. Tret’ego Vsesoyuz. Mat. S’ezda. T. 1 (Proc. Third All-Union. Math. Congr. Vol. 1), Moscow, 1956, pp. 65–66.
[7] Aldashev, S.A., Nonuniqueness of the solution of the spatial Gellerstedt problem for the multidimensional Lavrent’ev–Bitsadze equation, in Mater. mezhdunar. konf. “Differentsial’nye uravneniya. Funktsional’nye prostranstva. Teorii priblizhenii”, posvyashch. 100-letiyu akad. S.L. Soboleva (Proc. Int. Conf. “Differential Equations. Functional spaces. Approximation Theory.” Dedicated to the 100th Anniversary of Acad. S.L. Sobolev), Novosibirsk, 2008, p. 93.
[8] Aldashev, S.A., Nonuniqueness of the solution of the spatial Gellerstedt problem for a class of multidimensional hyperbolic-elliptic equations, Ukr. Mat. Zh., 2010, vol. 62, no. 2, pp. 265–269. DOI: https://doi.org/10.1007/s11253-010-0352-4
[9] Moiseev, E.I., Nefedov, P.Kh., and Kholomeeva, A.A., Analogs of the Tricomi and Frankl problems for the Lavrent’ev–Bitsadze equation in three-dimensional domains, Differ. Equations, 2014, vol. 50, no. 12, pp. 1677–1680. DOI: https://doi.org/10.1134/S001226611412012X
[10] Mikhlin, S.G., Mnogomernye singulyarnye integraly i integral’nye uravneniya (Multidimensional Singular Integrals and Integral Equations), Moscow: Gos. Izd. Fiz.-Mat. Lit., 1962.
[11] Kamke, E.W.H., Differentialgleichungen, Leipzig: Teubner, 1959. Translated under the title: Spravochnik po obyknovennym differentsial’nym uravneniyam, Moscow: Nauka, 1965.
[12] Kolmogorov, A.N., Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (The Elements of the Theory of Functions and of the Functional Analysis), Moscow: Nauka, 1976.
[13] Smirnov, V.I., Kurs vysshei matematiki (A Course of Higher Mathematics), vol.4, part 2, Moscow: Nauka, 1981.
[14] Bateman, G., Erdelyi, A. Higher Transcendental Functions (Russian translation), vol. 1, Moscow: Nauka, 1973.
[15] Sobolev, S.L., Nekotorye primenenija funktsional?nogo analiza v matematicheskoj fizike (Some Applications of the Functional Analysis in Mathematical Physics), Novosibirsk, Izd. SO AN SSSR, 1962.
[16] Aldashev, S.A., Some local and nonlocal boundary value problems for the wave equation, Differ. Equations, 1983, vol. 19, no. 1, pp. 3–8.
[17] Aldashev, S.A., Kraevye zadachi dlja mnogomernyh giperbolicheskih i smeshannyh uravnenij (Boundary-Value Problems for Multidimensional Hyperbolic and Mixed Equations), Almaty: Gylym, 1994.
[18] Aldashev, S.A., On some boundary value problems for a multidimensional wave equation, Dokl. Akad. Nauk SSSR, 1982, vol. 265, no. 6, pp. 1289–1292.
Additional Files
Published
Issue
Section
License
Copyright (c) 2024 Kazakh Mathematical Journal
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
One can find the license terms "CC Attribution-NonCommercial-NoDerivatives 4.0" here.
